The irreducible characters of the simple group of M. Suzuki of order 448, 345, 497, 600

JOURNAL OF ALGEBRA 29, 303-323 (1974)

The Irreducible Characters of the Simple Group of M. Suzuki of Order 448, 345,497, 600

DONALD WRIGHT

Department of Mathematics, University of Texas, Austin, Texas 78712*

Communicated by W. Feit

Received March 7, 1972

The objective of this paper is to obtain the character table of SZ, the simple group of order 213 . 3’ . 52 * 7 . 11 . 13 discovered by M. Suzuki [3, p. 1131. The character table of G,(4), the Chevalley group of type (Gz) over the Galois field GF(4) of four elements, will also be obtained.

The simple group G,(4) of order 212 . 33 * 52 . 7 . 13 was originally discovered by L. E. Dickson [6, 71 and reappeared among the groups discovered by Chevalley [4] in 1955. A complete description of the Chevalley groups of type (G,) can be found in [20].

It has been shown by Suzuki and separately by Wales [24] that the Hall- Janko simple group HJ of order 2’ . 33 . 52 . 7 is a subgroup of G,(4). In building the character table of G,(4) use will be made of the character table of HJ built by Wales and Hall [14] with the aid of computer, and their notation will be adhered to. In [3, p. 1131, Suzuki defined SZ as a primitive transitive extension of degree 1782 of G,(4).

Throughout this paper HlK will denote an extension of a group H by a group K, C,, the cyclic group of order n and W, the elementary abelian group of order n. Finally, if k, , K, , k, are three elements of SZ, then c(K, , K, , K3) is defined to be the number of solutions (x, y) in 5’2 of the equation x . y = k3 with x N k, in SZ and y N k, in SZ.

THE CHARACTER TABLE OF G,(4)

G,(4) acts as a primitive permutation group on the 416 left cosets of HJ; in the action on 416 points HJ fixes a point and has two further orbits (see [3, p. 1131) whose lengths can only be 100 and 315.

In this section C(x), N(x) will denote the centralizer, normalizer of any subset x of G,(4); also if f is any character of HJ then .$t denotes the character obtained by inducing [ to G,(4).

* Present address: Department of Pure Mathematics, University College of Wales, Penglais, Aberystwyth SY 23 3B2, U.K.

303 Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.

304 DONALD WRIGHT

The permutation character of HJ acting on 100 points is Ys + Yi -i- Ys (see [14]) while on 315 points it is Y,, + Yi + Ys + Ya + Yi, + Yru,, . Let x be the permutation character of G,(4) acting on 416 points; then xJHJ = 3Y,, + 2Y, + Ya + Ys + Y, + Yr4 + Y1, and its values on the classes of H J are as follows.

X e J II RK T 44

/C,(x)1 27.331.5P.7 2’?5 24.:.5 2543 8 3 23 23 . 33 . 5

X&J 32 16 4

X Tl TJ TlJ1 TR n n2 rr 44 3 6 6 12 5 5 5

1 C,,(x)/ 2s * 3s 23 * 3 22 . 3 22 . 3 2s * 3 * 52 22 . 3 . 5s 2s . 3 * 5s

XlHJ 5 8 1 4 1 1 6

X fl12 nJ1 n2Jx GJ n;‘J =T I12T a 44 5 10 10 10 10 15 15 7

, CIfJ(X)l 23-3.52 22’ 5 22.5 2.5 2.5 3 *5 3.5 7

dH.I 6 1 1 2 2 1 1 3

One immediately sees that there is no coalescence of these classes in G,(4), and that the centralizers of these elements in G,(4) have the following orders.

X e J JI RK T 44 1 2 2 4 8 3

I w 212 . 33 . 52 . 7 . 13 212 . 3 . 5 28 . 3 . 5 29 . 3 25 26 . 33 . 5 . 7

X Tl TJ TIJI TR = n-2 4 44

i2 266 224 24123 5 5 5

[ C(x)1 22 . . 5 3 3 22 . 3 . 52 22 . 3 . 52 22 . 3 . 52

X a2 I;rJ1 f12J, 4J 4”J IlT I12T a 44 10 10 10 10 15 15 7

1 C(x)1 2s * : . 52 22’5 22.5 22.5 22.5 3 -5 3 -5 3.7

x is, of course, zero on all remaining classes of G,(4). Now G,(4) has precisely two classes of involutions (see [23]); C(J) is isomorphic to an extension of a group of order 21° by A, while C( J1) is isomorphic to an extension of a group of order 26 by A,. It is proved in [14] that C&T)/(T) is isomorphic to A,; since, A, and L,(4) are the only simple groups of order

SIMPLE GROUP OF SUZUKI 305

+(8 !) (see, for example, [9, Appendix A]) it follows that C(T)/(T) is isomorphic to L,(4).

Now L,(4) has a nonreal pair of classes of elements of order seven; hence, N(( T)) has a single class of elements of order seven and G,(4) has a real pair of classes of self-centralizing elements of order twenty-one, with representatives UT and aT2. Also there are two further classes of elements of order twelve associated with T; each fixes none of the 416 points and is centralized by a group of order 2* * 3 in G,(4).

C( TJ/( TJ has self-centralizing elementary abelian Sylow 2-subgroups of order four; application of [12, p. 421, Theorem 2.11 yields that

Hence, there are two further classes of elements of order fifteen which are real and self-centralizing, with representatives 17,T, , D12Tl . Let b be an element of order thirteen of G,(4); then b centralizes no elements of order two, three, five, nor seven and application of a theorem of Sylow yields that G,(4) has a pair of nonreal classes of self-centralizing elements of order thirteen. On counting the elements in the classes obtained so far one finds that there still remain 23 * 32 . 52 * 7 * 13 . 53 elements to be accounted for; these must all be elements of 2-power order.

The permutation representation of G,(4) on 416 points is rank three and the degrees of the transitive constituents of H/ are 1, 100 and 315. A theorem of Frame [26, p. 891 yields that the degrees of the irreducible constituents of the permutation representation are 1, 65 and 350. Let x0 be the trivial character of G,(4), x1 the constituent of degree 65 and x2 that of degree 350. Then

x = x0 + x1 + x2 , X~HJ = U, + YI + y14 + Flu,, and

x2h = yo + Yl + y2 + ‘iv, + y, *

A result to be found in [5] asserts that G,(4) has two irreducible characters of degree (4/3)(4* + 42 + 1) = 364, both of which are rational on the whole group. Let these be x3 , x4 . Also the Steinberg character (see [5]) of G,(4) is irreducible, rational and its degree divides 1 S / = 212, S being a Sylow 2-subgroup of G,(4).

One may now apply some results of Brauer [l] as follows. There is one 13-block Bo(13) of defect one consisting of six irreducible characters whose degrees are congruent to *l (mod 13) and a complex conjugate pair whose degree is congruent to f6 (mod 13). M oreover there are two 7-blocks of defect one; the principal 7-block B,(7) consists of seven irreducible characters whose degrees are congruent to 51 (mod 7) while the 7-block B,(7) consists of three irreducible characters whose degrees are congruent to f2 (mod 7) and a real pair whose degree is congruent to f 1 (mod 7). All irreducible characters of G,(4) outside these three blocks have degrees divisible by 7 * 13 ;

306 DONALD WRIGHT

also one sees immediately that the Steinberg character has degree 4096 or 64. But if G,(4) had a rational irreducible character of degree 64, its restriction to HJ would be Yr + Yi, + Yn by Frobenius’ reciprocity theorem, so its value on T would be 19 = -2(7), contradicting Theorem 9 of [I]. Hence, the Steinberg character x5 has degree 4096; with a little effort it is now possible to evaluate x5 on the whole group; this proves to be extremely useful as the construction of the table progresses.

One already has x0 , x2 , x5 E Bs(13). With some effort it is now possible to show that the degrees of the remaining characters in B,,(13) can only be 4725, 4725, 378, 300 and 300. Let the corresponding characters be xa , x, , xs , xs , and xi,, . It is worth noting that x6 , x7 , and xs are of full 3-defect. Now xs, xl,-, may be evaluated on all known classes and it is seen that xsJHJ = xlsJHJ = Us; also xs, x7 are of full 5-defect and may be evaluated on all known classes. Now one obtains xsJHJ = Yn, + Ya, .

One already has x0, x5 , xs , and xi0 E B,(7). The degrees of the remaining characters of B,(7) can only be 78, 650 and 2925. Let xl1 , xra , and xi3 be the corresponding characters. One obtains xlrJHJ = Yi + Yrs + Yr, , xi&.,, = ul, + y3 + ‘iv, + Yll , and xi3 may now be evaluated on all known classes. One already has xi E B,(7); after applying Brauer’s conditions and considering possible restrictions to HJ, seven possible completions of the degrees of characters in B,(7) are obtained. However, by attempting to split Yrdt, of length five, and then reconciling each case with the inner products of Yiu,,t and x1 . xi1 and second-level tensor powers of x1 and xi1 , a contradiction may be obtained in every case but three. It is now convenient to split x[121JHJ into irreducible characters of HJ; on attempting to decompose Y,,f and ~~“1, noting that (Yi4f, &?I) = 4, it is found that only one case remains. This gives xl4 , xrs , xrs , and x1, of degrees 2925,2925,4160, and 1300, respectively. With some effort it is now possible to complete the evaluation of all the irreducible characters found so far on all the classes already obtained.

When an irreducible rational character of degree 364 of G,(4) is restricted to HJ, there are only two possibilities which can occur; one is Y, + Us + Ys + Yn, + Y1, + Yrs and the other Ya + Us + Y,, . The above calcula- tions showed that both necessarily occurred; let x3 correspond to the former and xa to the latter. By considering various inner products between the four characters Yiu,t, Yst, Y,,t and ~~~~1 and restricting Yrt - x1 - xs - x3 -

x5 - x11 - x13 - Xl6 - x17 ) of length two, to HJ and decomposing into irreducible constituents, three irrational pairs of irreducible characters arise. These are xls , xi9 of degree 819, xzO , xzl of degree 3276 and xz2 , ~33 of

degree 4095. The TR-column may now be filled by inserting the value -1 for an

irreducible character x24 of degree 1365, which can be evaluated on all known classes purely by column orthogonality relations. Similarly the ITT-column


may now be completed by inserting 1 for the value of an irreducible character

x25 of degree 5460, which can be evaluated. The T-column has now been completed; hence each remaining degree is divisible by 3 * 7 * 13.

By considering inner products between !P2r, Y,t, YIst, and ?PI,t and other information accumulated so far, one obtains three more pairs of irreducible irrational characters; Y2, , Y2, have degree 819, x2s , ,v2s have degree 4095 and x3,, , xsr have degree 3276. This final pair is obtained by considering the sum of the squares in the e-column, and Yat - x2 - x5 - xl2 - xl3 - x14 - 2x1, - x20 - x21 - X24 - x23 - x29 = x30 + x31.

Hence, G,(4) has precisely 32 conjugacy classes; of these 29 have been found. The remaining three consist of 23 . 32 . 52 . 7 . 13 . 53 elements of %-power orders. Elementary computation shows that there are two possibilities for the orders of the centralizers of the elements of the three remaining classes; these are (a) 25, 2s . 3, 2’ . 3 and (b) 25, 28 . 3, 2s. Further computation shows that xs cannot be evaluated on the three classes in case (a) such that it will have length one. Hence, case (b) holds. Let Ii, be an element of order four such that 1 C(R,)I = 28 . 3 ; then TR, and TR;’ are representatives of the two classes of elements of order twelve mentioned previously. Let R, be an element of G,(4) such that 1 C(R,)j = 2s, and let KI be a representative of the final class, so [ C(K,)/ = 25. I n order to ascertain the orders of R, and Kl a discussion of the Sylow 2-subgroup S of G,(4) is now necessary. This may be found in [23] but is reproduced here because knowledge of S will be of some importance in a later section. The full theory can be found in [20] and [16].

Let A be a simple Lie algebra of type (G,) over the complex field and let Z be the set of roots of A. Relative to some ordering on Z, supposed fixed from now on, the fundamental roots of A may be denoted by a and b with b > a. The set of positive roots of A is then .Z+ = {a, b, a + b, 2a + b, 3a + b, 3u + 2b) and the set Z consists of the elements of Z+ and their negatives. G,(4) is the Chevalley group of type (G,) over GF(4) so it has the following properties.

(i) For each r E Zf there is a (nontrivial) homomorphism

$6 sL2W -+ G2W For each (I! E r define

Note that w, and each x,(a), OL # 0, are involutions.

(ii) Let S, = {x7(a): OL E I’}. Each S, is an elementary abelian group of order four with multiplication defined by the formula x,(a) x,(,3) = x,(or + /3), where a, p E r.

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(iii) S = S S S S S S a b a+b 2a+b 3a+b Sa+2b has order 212 so is a Sylow 2-subgroup of G,(4).

Any element x E S can be expressed in exactly one way as a product of the

form x = %(%) xb(%z) %+b(‘%) X2a+bk%Y4) X2a+b(ar5) X2a+2b(%s) where each

01~ E r. The rules for multiplying any two elements of S are as follows; they are obtained by reducing certain results of Ree [ZO] modulo 2.

iXab>, xb(81 = %+bh+) X2a+b(a2p) X2a+b(a3f%

[xa(a>, xa+b(p)l = %a+bta28 X3a+2b(a~2>,

[Xa(a)> X2a+b(fl)1 = %+bh%

[Xb(a>, X2a+b(fi)1 = %a+2bh%

[Xa+b(ol)> X2a+b(r6)l = %+2b(‘&

For all other pairs of positive roots r, s one has [~~(a), x&3)] = 1. An inspection of these results now shows that R, has order four while Kl

has order eight. Also one notes that representatives of the G,(4)-classes may be chosen such that R2 = RI2 = R22 = J, K2 = R and K,2 = R, . The remainder of the power map is obvious from the notation; in particular one notes that the involution J1 is square-free. Routine use of orthogonality relations, etc., now enables the character table of G,(4) to be completed, as shown in Table I.

THE CHARACTER TABLE OF SZ

SZ acts as a primitive permutation group on the 1782 left cosets of G,(4); in the action on 1782 points G,(4) fi xes a point and has two further orbits whose lengths are 416 and 1365.

In order to build the character table of SZ it is found necessary to make use of two further subgroups of SZ. These are the centralizer in SZ of the group (J} of order two and the normalizer in SZ of the group (T) of order three; these will be discussed later. In this section C(X), N(x) will denote the centralizer, normalizer respectively of any subset x of SZ. If 6 is any character of SZ, 51 will denote the character of G,(4) obtained by restricting 6, while if 17 is any character of any subgroup of SZ, 71 will denote the character of SZ obtained by inducing 7.

The permutation character of G,(4) acting on 416 points is x0 + x1 + x2 , while on 1365 points it is easy to see that it must be x0 + x2 + xs + xl2 . Let Z be the permutation character of SZ acting on 1782 points; then ZJ = 3x0 + x1 + 2x2 + x3 + xl2 and its values on the classes of G,(4) are as follows.

481/29!2-8

310 DONALD WRIGHT

X e J J1 R 4 4 K & T Tl TJ TIJ, TR TR, 44 1 2244488336 6 12 12

a 1782 54 42 30 2 6 6 2 162 18 18 6 6 2

X TR;l II IP l& G2 17J1 n2J, =,J 4’J o(x) 12 5 5 5 5 10 10 10 10

2-1 2 2 2 12 12 2 2 4 4

X 17T l12T IIIT, II12TI a aT aT2 b b-l o(x) 15 15 15 15 7 21 21 13 13

ZJ 2 2 3 3 4 1 1 1 1

One immediately deduces the following orders of centralizers of elements in SZ; questions regarding coalescence in SZ of the remaining G,(4)-classes will be answered later.

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1 J J1 R 2 2 4

I WI 213 . 37 . 52 . 7 . 11 . 13 213 . 34 . 5 29 . 32 . 5 . 7 210 . 32 . 5

X RI R2 K 4 T Tl TJ 064 4 4 8 8 3 6 6

I C(x)1 29 . 3 2ra . 3 26 . 3 26 27 . 37 . 5 . 7 23 . 34 . 5 27 . 33

X T~JI TR lI,T, lI12TI aT aT2 b b-l 44 6 12 15 15 7” 21 21 13 13

[ C(x)] 23 * 32 25 * 32 32 * 5 32 * 5 22 . 3 . 7 3 * 7 3 . 7 13 13

SZ has precisely two conjugacy classes of involutions; for if it had an involution J2 conjugate neither to J nor to J1 , then J2 would fix none of the 1782 points and hence SZ would not be a subgroup of A,,,, , a contradiction. Let c be an element of order eleven in SZ; then c centralizes no element of order five, seven, thirteen, nor two and application of a theorem of Sylow now yields that SZ has a single class of self-centralizing elements of order eleven. The following has been proved.

LEMMA. SZ has precisely two conjugacy classes of involutions. SZ has a single class of self-centralizing elements of order eleven and a nonreal pair of classes of self-centralizing elements of order thirteen.

The next result determines the classes of elements of order five of SZ,


LEMMA. SZ has precisely two conjugacy classes of elements of order jive with representatives 17 and I71 . C(D) N (l7) x A, while C(l7,) N (l7,) x A,.

The proof of this lemma follows. In the action on 1782 points, HJ fixes two points. The involution inter-

changing these two points normalizes HJ but certainly does not centralize HJ (see [3, p. 1131). In [14] t i is shown that HJ has an outer automorphism (II of order two which together with the inner automorphisms gives all automorphisms of HJ. Moreover, such an 01 is exhibited together with various other elements of HJ as elements of $ioo . Using the character table of HJ to ensure a maximum probability of success, an element n is found from the given elements, and 17 * (n2p found to be an element of order twelve, but a computation using the character table of HJ shows that no product of a conjugate of n with a conjugate of D2 can have order twelve. Hence, I72 N n in Aut(HJ). Similarly it may be shown that LY fuses the elements 171 , l7,2.

Hence, 1 C(n)1 = 22 * 3 . 52 and C(D) N (n) x A, . So 17JI - I72 J1 in SZ, 1 C(17J,)I = 22 * 5, and l7T N n2T in SZ, 1 C(nT)I = 3 . 5. Also 1 C(fli2)l = 23 . 32 * 52 and Di J - U12J with 1 C(17, J)I = 23 * 5. Now in G,(4), 17i is centralized by a group isomorphic to (n,) x A, . Consider G,, = C(17,)/(II,), of order 23 . 32 . 5; a Sylow 2-subgroup of G, has an element of order four because 5 divides j C(R)I. Suppose it is abelian; application of a result of Walter [25] yields that O’(G,/O(G,)) ~1? C, x A, , this being impossible since then G,, would have elementary abelian Sylow 2-subgroups. Hence a Sylow 2-subgroup of G, is dihedral; application of a result of Gorenstein and Walter [13] yields that G,,/O(G,) E A, or $5 according as I O(G,)I = I or 3. Suppose / O(G,,)/ = 3; a well-known result of Schur implies that G, N_ $5 x C, , so that in (17,)\G, one may choose an element of order three which centralizes a Sylow 5-subgroup of C(n,) and hence an element 17. Now 3 divides exactly 1 C(n)1 so this element of order three can only be conjugate to T in SZ; but 5 divides exactly / C(T)I, so one has a contradiction. Therefore / O(G,)/ = 1, G N A, and C(I;T,) E

X4> x A,. Hence, C(17,) has elementary abelian Sylow 3-subgroups which consist

entirely of elements conjugate to TI in SZ. There is a class of elements of order twenty in SZ with representative fl,R and 1 C(n,R)I = 22 * 5. The Centralizer of an Involution J in SZ will now be considered.

LEMMA. The group C(J) is isomorphic to an extension of a group E of order 27 by O,(3) ; the group E is extraspecial and possesses precisely 54 noncentral involutions. Also Z(E) = Z(H) = (J>.

Let H = C(J) and R = H/(J). So ) i7 ) = 2l2 * 34 . 5 and by a result

312 DONALD WRIGHT

of Brauer [2] fi is not simple. From a previous remark it is known that A, is involved in f7. Suppose R has a normal non-abelian simple subgroup; then by [2] this is a group M isomorphic to A, , A, or O,(3).

(i) Suppose M N A,; then 1 Aut(M)/M 1 = 2 and 1 M 1 = 22 - 3 * 5. Now (L$) is a Sylow 5-subgroup of H; hence, l7r would have to be centralized in SZ by a group whose order is 2” where x > 4, which is not the case.

(ii) Suppose M c1 A, so that 1 Aut(M)/M 1 = 4 and 1 M 1 = 23 - 32 * 5. Again an element J7, in SZ would be centralized by a group of order 2” where x > 4, a contradiction. A similar contradiction is obtained if M N O,(3), for 1 Aut(M)/M I = 2 and 1 M 1 = 2s * 34 - 5.

Hence, it may be assume that fl has a normal p-subgroup where p = 2 or 3. Suppose H has a normal 3-subgroup; this must have order 3, 32, or 3s. Then an element Lrl centralizes in SZ an element of 3-power order and J which commute. This contradicts the structure of C(17,). So it may be assumed that R has a normal 2-subgroup P. An element n1 of fl cannot centralize a subgroup of P of order greater than 22, so the only possible orders for P are 2, 22, 2*, 25, 26, 2s, 29, 21°. It is now easy to see that O,(3) is involved in H; for Aut(AJ E $s , 1 Aut(A,)I = 25 . 32 . 5, A, has Schur multiplier two and A, has Schur multiplier six, while SZ has no element of order thirty.

Now the two smallest degrees of permutation representations of O,(3) are 27 and 36 (see [8, p. 3071) ; so either P is centralized by a group MO tz O,(3) or 1 P / = 26. Hence, P has order 2, 22, or 2s. If 1 P I = 2 or 22 then by applying the above arguments to f7/P one sees that f7 must have a normal subgroup of order 26. So I P I = 26 and R/P N O,(3). Thus, H is an extension of a group E of order 2’ by a group MO N O,(3). The following conjugacy classes of O,(3) are due to Frame.

Class Order of element Order of centralizer

1 2 3 4 5 6 1 2 2 4 4 3

in O,(3) 26 . 34 . 5 25 . 3 26 . 32 24 . 3 23 22 . 33

rl 27 7 3 3 1 9 5 36 8 12 0 2 6

Class 7 8 9 10 10-l 12 12-l Order of element 6 6 3 6 6 3 3 Order of centralizer

in O,(3) 2 . 32 22 . 3 2 . 33 22 . 32 22 . 32 23 . 34 23 . 34

77 0 0 0

1 3 0 0


Class 14 14-l 16 16-l 18 18-l 20

Order of element 6 6 12 12 9 9 5 Order of centralizer

in O,(3) 23 * 32 25 . 32 22 ’ 3 22 . 3 32 32 5

77 0 0 2 5 0 0 1

Here the classes of elements associated with a particular class of elements of order three immediately follow that class. Now H = E\M, where E has order 27 and M,, N O,(3) ; M, has an action on E and must have the following orbits. M, fixes two elements, viz. e and J, and permutes the remaining 126 elements in four orbits of lengths 27, 27, 36, 36. It is easy to find the permutation characters of O,(3) acting on 27 and 36 points, using the character table due to Frame. It is found that in the action of M,, on E an element of class 6 fixes 25 points, an element of class 9 fixes 23 points, elements from classes 12, 12-l fix 2 points and elements from classes 18, 18-l fix 2 points. An element of class 20 fixes, of course, 23 points. It is known that H has an element of order three conjugate to T in SZ and centralized in H by a group of order 2’ . 33; this must correspond to class 6.

Corresponding to class 9, there is an element T3 of SZ which centralizes / such that [ C(T3J)[ = 2* * 33 and T3 + Tl in SZ. Corresponding to classes 12, 12-l there are two conjugacy classes of elements of order six of SZ with representatives T, J, Ts2 J such that 1 C(T, J)I = 1 C( Ts2 J)I = 2* . 34 where Tz is an element of order three of SZ and T, + Tl in SZ. Finally, corresponding to classes 18, 18-l there are two conjugacy classes of elements of order eighteen in SZ with representatives U J, U-l J such that j C( U J)I = 1 C( U-lJ)l = 2 . 32, where U is an element of order nine in SZ. Note that H has no conjugate in SZ of Tl .

The structure of the group E will now be obtained. A Sylow 3-subgroup of M,, has order 3* and acts faithfully on E; hence, [ D(E)] = 2. So D(E) = E’ = Z(E) = (J), each group having order two, and E is an extraspecial group of order 2’. If E is isomorphic to the central product of a dihedral group of order eight with two quaternion groups, then E has 56 elements of order four and M,, cannot have the required orbits on E. Hence, E is isomorphic to the central product of three quaternion groups. The group E has 54 noncentral involutions and 72 elements of order four. If x is any noncentral element of E then [E: C,(x)] = 2, and the conjugates of x in E are x and x J.

It has been seen that Z(S) = S3=+s0. Let xga+sb( 1) be the central involution of E, the field r consisting of the elements 0, 1, p, ps. Then x~~+&P) N ~~~+s~(ps) in E, and H has a conjugacy class consisting of precisely 54 involutions with

314 DONALD WRIGHT

representative ++s&). Now Z(S) char S Q S implies that Z(S) Q S, where S is a Sylow 2-subgroup of H containing S; so if k is an element of S - S, then x~~+~~(P)” = x3a+2b(~2).

THE CENTRALIZER OF AN ELEMENT T OF ORDER THREE IN 5'2

LEMMA. The group C(T) is isomorphic to an extension of (T) by U,(3).

It is known that the centralizer of T in G,(4) is isomorphic to an extension of (T) by L,(4). Let X = C(T)/(T) so that L,(4) < X and

[Xl =2’~3~.5-7.

Certainly / O(X)1 is divisible neither by 5 nor 7. Suppose / O(X)/ # 1; then O(X) has order 3, 32, 33, or 34. But now it is impossible for a Sylow 5-subgroup of X and a Sylow 7-subgroup of X to act on O(X) without fixing some element; this is impossible since 3 divides exactly 1 C(n)\ and 1 C(a)l. So 1 O(X)1 = 1 and clearly 1 O,(X)1 = 1.

Now suppose X is not simple; then X has a proper normal non-abelian simple subgroup Y and Y has a subgroup isomorphic to L,(4). Hence, IYI =2”*3b*5*7where6<a<7and2<b<5;inanycase there is an element of order three which centralizes an element of order five in X. This is impossible because 3 divides exactly 1 C(n)l. Hence, X is a simple group.

Attention will now be focused on a Sylow 2-subgroup of X, of order 2’. A Sylow 2-subgroup Fi of the centralizer in G,(4) of T is isomorphic to SbS2a+bS2a+2b; Fl is a special 2-group of order 26 each of whose involutions is conjugate in G,(4) to J, and Z(F,) = Ssa+ab . Also Fl has precisely two elementary abelian subgroups of order sixteen, viz. SbSsa+sb and Ssa+bSsa+sb . Now an element K of order eight is centralized in SZ by some element of order three which must be conjugate in C(J) to T. A Sylow 2-subgroup F of X is isomorphic to S b S 3a+bS3a+2bC2; let k be some element of order eight of F whose square lies in SbS3a+bSsa+2b. It may be assumed that

k2 = xb(d x2a+b(~2) %a+2b(P2)*

So k centralizes xb(p) X2a+b(P2) X2a+2b(P2)y sends X2a+2b(P) to X2a+2b(P2) under

conjugation and inverts xb(p) x aa+b(~2). Computation now shows that F is isomorphic to an extension of (al , a2 , b 1 , b * aI2 = a22 = b12 = b22 = t, # 1, 2. to2 = 1, b;‘alb, = a ;l, bg’a b - 2 2 - ai’, ala2 = a,a, , a,b, = b,a, , a,b, = b,a,)


by (u, V) where (u, v) is dihedral of order eight and v-iuiv = a;‘, V-%Z~V = u;i, v2 = t, , v-%,v = bra, , v-%,v = bsa, . The isomorphism is now exhibited.

al- %(P) %+bw a,+--+ %(P) %7+dP2) %a+2tiP2h

b1- %a+b(P2) %a+zb(l)4)k v - f%(l) %,bU>l

b2- %a+b(l) %(l)k 24 f-) x*( 1).

Hence, F is isomorphic to a Sylow 2-subgroup of U,(3) (see [18, p. 791). It has been shown that X is a simple group of order 27 * 36 . 5 * 7 with Sylow 2-subgroup F. A result of Harada [15] now implies that C,(l) is isomorphic to the centralizer of some involution in one of the simple groups A,, , Ms2, M2s, HI, MC, L,(3), L,(5), or U,(3), where (J) = Z(F). A result of Phan [19] implies that Cx(J) is not isomorphic to the centralizer of any involution in L,(3) nor L,(5), for I X j # 1 L,(3)1 and I X I # I L,(5)l. None of the groups A,, , M,, , M2s , HJ, M c h as an involution centralized in that group by a group of order 27 . 32 = I Cx(J)/. Hence, C,(f) is isomorphic to the centralizer of some involution in U,(3). Another result of Phan [18] now implies that X N U,(3) ; h ence C(T) E (T)\U,(3). The character table of U,(3) has been constructed by Todd and is shown in Table II. The normalizer of (T) in SZ will be considered next.

Before the structure of N(( T)) is obtained it is worth noting that the group C(a)/(a) is non-abelian (because all involutions of C(T) are conjugate in SZ to J) and so one has the following.

LEMMA. The group C(a) is isomorphic to (a) x A, .

Let X = N((T))/(T); then X is isomorphic to some extension U,(3) of U,(3) by C2 . Since Aut( U,(3))/ U,(3) is isomorphic to the dihedral group of order eight, there are up to isomorphism three extensions of U,(3) by C, . Since an element aT of order twenty-one is conjugate to its inverse in SZ, some element of U,(3) must invert an element of order seven of U,(3). In only one of the three above-mentioned extensions does an extending element fuse the inverse pair of classes of elements of order seven of U,(3) and at the same time give a suitable number of involutions of U,(3) not centralizing T; this corresponds to an extension U,(3) of U,(3) which is not the extension by the involutary field automorphism.

Note that TR, = TR;l in SZ and that I C(TR,)I = 2* * 3. Also 1 C(TK)I = 23 .3, TKb em ’ g an element of order twenty-four. The conjugacy classes of U,(3) have been determined by Finkelstein [9, Table VI]. They are as follows.

3, 3, 61 62

12, 5 7

28 * 3s * 5 * 7 28 - 32 26 - 3

25 24

24 * 36 22 ’ 36 2 * 34

24 * 32 22 - 32 23 * 3 2.5

7

22

4, 82(4J

83 84

64(3,)

24, 10

W2)

9;l

25 * 32 ’ 5 25 - 3

23

25 ’ 3 25

2 - 32 22 ’ 3 23 ’ 3

23 - 3 2.5

33

33

1 2 41 41, * 3, 3', 3"' jJ% g' g" $', 9"" 6' 6" 6"1 ,2 5 ,, 7,1

2?3! 7 2 5 5,7 2 3 2.3 24 23 23362’35 2!35 34 3’ 33 3= 33 ,233’ 2?32 2f3= 213 5 7 7

1 1, 1 11111 1 11 1 11111 11

21 5 1 1 -1 -63 3 3 3 - - . . 2 -1 -1 -2 1 . .

169-3 5 1 127. , * . . . . 3 . . -1 -1 . .

896 . . . 32 -4 --4 -4 -1 -1 -I -1 . . . . , . .

7299-3 ,-, , . . _ , - - - . .-j 1 1

90 IO-2 2 . 9 9 9 - . I . - 1 1 1 1 . -1 -1

640. - . - -8-8-8 , 1 1 1 1 . - - - . g 2

640. . . e-8-8-8 1 1 1 1 I . . . , _ 2 o(

140 12 4 . . 5 -4-4 5 -1 -1 -1 -1 -3 . - 1 . . _

210 2 -2 -2 21 3 3 3 . . . 5-l-l 1 . . -

560 16 . . . -34 2 2 2 -, -1 -1 -1 2 2 2 . _ ,

315 11 -1 -, 1 -9 18 --9 . . . , , -1 2 -1 -1 . . -

35 3 3-l-l 8 6--1--1-l-l 2 2 . . 3 . , . .

315 11 -1 -1 l-9 -9 16 . . . , I -, -1 2-l...

35 3 3 -1 -1 8 --1 8-l 2 2-l-l . 3 . . . . -

420 4 4 , .-39 6 6-3. . . . l-2-2 1 e - _

28+6 . # - 10 1 10 1 /+37"3,/ 1 l-2 l-2 . . . .

260-6 1 . 10 , (0 1 it3p ,+qz 1 1 -2 1-2 I . , .

260 -8 * , 10 10 1 1 1 , ,+3~/+3p -2 1 - . . I

280-E . . . 10 10 1 , , 1 ,++ '"7 -2 -2 1 . . . .

,0=-4(-1+~~ (* = %(-I+ "-)

THliE TT PSU,lJ’J


Each element of U,(3) is a coset of (T) in N((T)). Each class in U,(3) of elements whose orders are not divisible by 3 gives rise trivially to three conjugacy classes of C(T) and the fusion of the latter under the action of an element which sends T to T2 is trivially determined. Each class in U,(3) - U,(3) gives rise trivially to a single conjugacy class of N((T)).

Now N((T)) contains a Sylow 3-subgroup of SZ, so an element Tl E N(( T)) ; it is known that a Sylow 3-subgroup of C( Tl) has order 3*, and, hence, an element 3s of m yields just one class of N((T)) consisting of precisely 27 * 32 * 5 * 7 conjugates of Tl . Also note that N((T)) has an element K, of order eight which is self-centralizing in N((T)) such that Ka2 = RI; since RI is square-free in G,(4), SZ has a class of elements of order eight which fix no points of the 1782. Moreover, K2* = J and so 1 C(K,)I is a power of 2.

By counting the number of elements of SZ already obtained, it is seen that a Sylow 3-subgroup of SZ has a center whose order is greater than three and that there is an element t in the center of some Sylow 3-subgroup which is not conjugate to T in SZ. One may assume that t E N(( T)) and is in the centre of some Sylow 3-subgroup of N(<T)). Since the 3,-coset must yield the conjugacy class containing t, t belongs to C(J). Hence t N T2 and/or t N Ta in SZ. The 3,-coset yields precisely two classes of N((T)); one consists of precisely 2* . 5 * 7 elements of order three conjugate in SZ on T, and the other consists of 25 . 5 * 7 conjugates in N(( T)) of t. Since tJ - t2 J in N((T)), t - T3 in SZ.

The 3,-coset must yield some conjugates in SZ of T, . If the 3,-coset yields precisely one class of N((T)) f 1 o e ements of order three, then T, J - T22 J in SZ, a contradiction. Also T, cannot be conjugate to T22 in M,(T)) for a similar reason. Hence, N((T)) h as a class consisting of precisely 26 . 3 * 5 * 7 conjugates of T, , a class consisting of precisely 26 * 3 * 5 * 7 conjugates of T22 and a third class of 26 * 3 * 5 * 7 elements of order three, one of which is T4 say; then T4 - T3 in SZ. Note that SZ has an element T,R of order twelve such that a Sylow 3-subgroup of C(T,R) has order 32, and also that 1 C(T,)I = 2” * 3’ where x > 4. Moreover, SZ has at most two more conjugacy classes of elements of order three besides those with representatives T, Tl , and T3.

The only elements of U,(3) which have failed to give complete information regarding the number of classes in N(( T)) are 9, and 9;l; note that !+s = 32 . An element U of N((T)) may be centralized in N((T)) by a group of order 33 or 3*.

One is now in a position to start studying the irreducible characters of SZ and at the same time to complete the determination of the conjugacy classes of SZ except for the elements of order nine and any 2-power elements which fix no points of the 1782.

318 DONALD WRIGHT

LEMMA. SZ has precisely three conjugacy classes of elements of order three with representatives T, Tl , and T3 . One has 1 C(T,)I = 24 ’ 37; moreover SZ is rankfive on N(( T)) and is rank eight on C(T).

The permutation representation of SZ on 1782 points is rank three and the degrees of the transitive constituents of G,(4) are 1,416, and 1365. A theorem of Frame [26, p. 891 yields that the degrees of the irreducible constituents of the permutation representation are 1, 780, and 1001. Let Z, be the trivial character of SZ, Zs the irreducible character of degree 780 and Za, the one of degree 1001, so that Z = Z,, + Zs + Z,, and Z,J = x0 + x1 + xz + x3 , z231 = x0 + x2+x12- On counting the elements of SZ obtained so far and attempting to induce various irreducible characters of C(T) to SZ, it may be seen that T, N T3 in SZ, 1 C(TJ = 24 . 3’ and / C(T,R)I = 23 . 32.

Let No be the trivial character of N((T)) and Nr the other character of degree one. The lengths of N,f and NIT may now be computed; more specifically, one obtains (Not, NoI) = 5 and (Nrt, NIT) = 3. Note that if Co is the trivial character of C(T) then COT = N,f + NIT. Also C(K,) is forced to be of order 25.

Using Brauer’s theory [l] of p-blocks of defect one for p = 7, 11, 13 the following results are obtained. There is one 13-block of defect one consisting of six irreducible characters whose degrees are congruent to f 1 (mod 13) and a complex conjugate pair of irreducible characters of degree congruent to &6 (mod 13). There is one 11-block of defect one consisting of eleven irreducible characters whose degrees are congruent to -+I (mod 11). Finally there are three 7-blocks of defect one; the principal 7-block b,(7) consists of seven irreducible characters whose degrees are congruent to &l (mod 7), the second 7-block b,(7) consists of seven irreducible characters whose degrees are congruent to 13 (mod 7), while the third 7-block b,(7) consists of three irreducible characters whose degrees are congruent to f2 (mod 7) and a real pair whose degree is congruent to f 1 (mod 7). All irreducible characters not belonging to any of these blocks have degrees divisible by 7 * 11 . 13.

Now NIT has length three and degree 22, 880; at this stage it is found necessary to embark on the tedious procedure of restricting Nrt to G,(4) and then decomposing into irreducible constituents of G,(4). Using the information and Brauer’s theory, detailed computation shows that

Nit = Z, + Z,, + Z,

where Z,(e) = 10725, Z,,(e) = 12012 and Z,(e) = 143; inner products between various induced characters from G,(4) to SZ were also used.


Here

and

a = x2 + x5 + x11 + x13 + x13 + x13 + x23 + x27,

z231 = x4 + Xl4 + x15 + Xl6 3

-a = x1+ x11*

An equally tedious procedure shows that N,f = Z,, + Zs + Z,, + Z,, + Z, , where Z,,(e) = 5940, Z,,(e) = 364 and Z,(3) = 15795. Also one has

'111 = x2 + x13 + x17 + x24 7

-u = x4 3

'91 = x2 + x4 + x5 + xl6 + x24 + x25.

Each irreducible character of U,(3) is an irreducible character of C(T), and when an irreducible character of C(T) is induced to N(T) the resulting character has length one or two. Using this and certain other information concerning evaluation of characters, the above irreducible characters may be evaluated on many SZ-classes of elements which fix no points of the 1782.

One already has Zs , Z, , Z,, , Z, E b,(7). Only one possibility exists for the degrees of the remaining characters in b,(7); this gives Z,, , Z,, , and Z,, of degrees 66,560; 193,050; and 248,832, respectively. In order to employ various tensor powers and products of characters, it is convenient to prove the following result.

LEMMA. SZ has an element R, of order four which$xes no points of the 1782, such that R32 = J1 . Moreover, R, and K, are the only elements of 2-power order whichjx nopoints of the 1782, and j C(R,)I = 25 . 32.

This result implies that the only ambiguity remaining in the determination of the conjugacy classes concerns the elements of order nine; to prove it, the group C(T,) of order 23 . 34 * 5 will now be considered.

It is known that the centralizer of 7’i in G,(4) is isomorphic to (Tr) x A, . Let C = C(T,)/(T,), so C has order 23 . 33 * 5; clearly O(C) has order 1, 3, or 32. Suppose first that a Sylow 2-subgroup of C is abelian; application of a result of Walter [25] yields that O’(C/O(C)) II C, x A, . Whatever the order of O(C), this gives a contradiction because an element Dr would have to centralize J1 in SZ (for no conjugate of Tl centralizes J). Hence, a Sylow 2-subgroup of C( Tl) is non-abelian; it contains at least three involutions so is isomorphic to the dihedral group of order eight. Hence, SZ has an element R, of order four such that R32 = J1 and one sees that C(T,R,) must have order

320 DONALD WRIGHT

2s . 32. By finding a few more irreducible characters of N( T), partly evaluating them and then restricting 2, to N(T) it can be shown that Z,(V) = 2 for any element V of order nine in SZ. Since 1 C(V)1 = 33, 3”, 2 . 33, or 2 . 34

and SZ has elements of order eighteen, one sees that Z,(R,) = -1, J C(R,)J = 25 . 32 and that all conjugacy classes of SZ besides those concerning the elements of order nine have been determined; moreover SZ has 213 * 34 . 52 * 7 * 11 * 13 elements of order nine.

It may be computed that Z, [“I - Z, has length two; restriction of Z$121 to G,(4) yields that Z, ~‘1 - Z, must be the sum of a pair of irreducible characters each of which has degree 5005. Let these be Z,, , Z,, . Again 24’1 - Z,, - Zs - Z, - Z,, is an irreducible character Z,, of degree 3432. One obtains

Z2,l = Z23l = x3 + x11 + x13 + Xl3 + Xl9

and

z141 = 2x1 + x3 + x17 + x13 + Xl9 *

Now consider x1? and xrrf of lengths nine and six, respectively; if the known irreducible characters contained in these are removed and the resulting characters restricted to G,(4) and split into irreducible constituents, then, using the fact that (xrt, xii?) = 2, sufficient information is obtained to produce four more irreducible characters Z,, , Z,, , Z,, , and Z,, such that Z,,(e) = 64064, Z,,(e) = 54054, Z,,(e) = 18954, and Z,,(e) = 25025. These satisfy

Xlt - z8 - z7 - 2214 = z39 + z~, + '32

and

Xllt - '1 - '7 - '27 - z23 = z39 + '33.

Now Z,, may be evaluated completely in a similar manner to that for Z, , and the scalar products (Z,, . Z, , Z,, * Z,) = 2, (Z,, * Z, , Z,,) = 1 computed. Hence, Z,, = Z,, * Z, - Z,, is irreducible of degree 40,040. Again Zs * Z,, has length three and has Z, , Z,, as constituents; hence, Z,, of degree 75,075 is obtained. By accumulating information concerning the relations between the 21 irreducible characters obtained so far and remarking in particular that Z,, , Z,, E b,(7), it is now possible to complete b,(7) uniquely; this gives Z, of degree 14,300; Z,, of degree 79,872; and Z,, , Z,, of degrees 64,350 (being a real pair). Using block orthogonality relations, etc., it is now possible to evaluate each irreducible character so far obtained on all classes, except of course the elements of orders nine and eighteen, with the additional exception that only the sums of certain pairs of characters are determined on certain classes.

TABL

E III

- e

r; 7

R,

R,

R

I, k;

K

Kx T

7;

T,

”

u-1 r

,r, l-

8 -r

,T T*

T’ T,

‘?‘T

,R, T,

.eI T

I T,

C 7

1, 7

-K U

I !P

z n;

1

ns, T

,T T

, pi T

f “,r

, Tr,‘

T D

~a

)\224

4448

8833

3996

6666

12

12

12 I

? 12

2,

ra

19

s 5

10 1

0 20

15

15 1

5 7

,cw

,~~~

y%

2T32

2Y,‘,

,3

3 *s

2”3

25

2731

34 4

7 3

3321

3,3,

44,2

23

2322

2 2

2 5,

,235

2.3

2.3

23 2

3 2.

3 23

23

23

23

23

25

P233

22’3

*3

3 23

2 23

23

52.3

5 25

T3

5 22

5 3.

5 x2

, 32

5223

1.

1 1

1 1

1 r,

‘072

5 -5

5 16

5 -1

:

45

.I I

I I

1 1

1 I

I 1

! I

I 1

1 :

A -I

6 1

1 1

1 1

1 1

1 1

I 3-

,2,.

I 1

-3

-?

-1

330

15

-2,

. b

-1

-s

3 3

, .

I.,

, ,

. I

I I

z*,6

e9m

.5,

-8

. .

. .

- .

-a,-,

5 I8

.

. .

. .

. 1

. .

s .

s ,

. .

--5--1

, .

1 .

1 z;

18

9540

36

-5

40

-12

-so

4 *

* ,

729.

.

. .

! 9

. .

I .

-3

. 1

--I

. .

. 5

1 s

-1

. :

I

2,

cl300

20

12

0 I2

-2

0 4

. -4

,

125

-,o

4,

-1

-, 2

I3

4 4

4 ’

I 4

1 --

I 1

1 1

%

1 .

, .

- -I

&-

,,s,y

164

-8

. .

. .

. I

,280

II

-16

2 2

-1

. ,

. I

- .

. -

. ,

-‘B--

J ,

. *

- ,

, .,

I‘ m

4949

0 -5

94

-18

5 2

-2

-2

a-72

9.

, *

. -

-9

. .

8 .

3 .

-, ,

* 1

9 ,

. 1

, ,

a,

143

-1

1s

-I -,

15

-1

3 -1

-1

35

-I 8

2 2

-, 3

6 6-

-‘l

2 3

-4-I

, d

8 3

-1

. .

. -1

-1

-4

2,

780

20

I2

* -4

28

4

- 4

. 10

5 6

-3

* *

* 9

-3

3 3

* -I

I 1

1 I

’ -

lb

’ -

2 -2

.

I 1

3 4

1579

5 55

24

3 1

3 27

15

-1

3

, 9

. .

, .

. .

3 ,

. .

. .

. .

1 -1

0 ’

- -2

2

d -,

-, 3

I,N88

32-5

, ,

8 .

. -

s .

. -9

.

: .

-, .

, .

, --

I .

. .

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322 DONALD WRIGHT

With the above information, a programme due to Guy was run at the Cambridge University computing centre and after some time yielded precisely eighteen more irreducible characters. Hence SZ has two conjugacy classes of elements of order nine with representatives U, U-l and 1 C(U)/ = 2 . 33. The completed character table is shown in Table III ; the power map for S.2 has been completely determined during the course of the construction.

ACKNOWLEDGMENTS

The author wishes to express his immense gratitude to Professor Donald Livingstone for his patience, guidance, and encouragement throughout the production of this work. The author is also indebted to the Science Research Council for financial support.

REFERENCES

1. R. BRAUER, On groups whose order contains a prime number to the first power, I, II, Amer. J. Math. 64 (1942), 401-420, 421440.

2. R. BRAUER, On simple groups of order 5 * 3” * 2b, Bull. Amer. Math. Sot. 74 (1968), 90&903.

3. R. BRAUER AND C. SAH, ED., “Theory of Finite Groups,” Benjamin, New York, 1969.

4. C. CHEVALLEY, Sur certains groupes simples, Tohoku Math. J. 7 (1955), 14-16. 5. C. CURTIS, Proceedings of Oxford Conferences on Finite Simple Groups 1969,

to be published. 6. L. E. DICKSON, Theory of linear groups in an arbitrary field, Trans. Amer. Math.

Sot. 2 (1901), 363-394. 7. L. E. DICKSON, A new system of simple groups, Math. Ann. 60 (1905), 137-150. 8. L. E. DICKSON, “Linear Groups,” Dover Publications, New York, 1958. 9. L. FINKELSTEIN, The maximal subgroups of Conway’s group C, and McLaughlin’s

group, Ph.D. thesis, University of Birmingham, 1970. 12. D. GORENSTEIN, “Finite Groups,” Harper and Row, New York, 1968. 13. D. GORJZNSTEIN AND J. WALTER, The characterization of finite groups with

dihedral Sylow 2-subgroups, I, II, III, J. Algebra 2 (1965), 85-151, 218-270, 334-393.

14. M. HALL AND D. WALES, The simple group of order 604800, J. Algebra 9 (1968), 417-450.

15. K. HARADA, Finite simple groups whose Sylow 2-subgroups are of order 2’, 1. Algebra 14 (1970), 376-404.

16. N. JACOBSON, “Lie Algebras,” Interscience Publishers, New York, 1961. 18. K. PHAN, A characterization of the finite simple group U,(3), J. Australian Math.

Sot. 10 (1969), 77-94. 19. K. PHAN, A characterization of four-dimensional unimodular groups, J. Algebra

15 (1970), 252-279. 20. R. REE, A family of simple groups associated with the simple Lie algebra of type

(G,), Amer. J. Math. 83 (1961), 432462. 23. G. THOMAS, A characterization of the groups G,(2”), J. Algebra 13 (1969), 87-l 18.


24. D. WALES, Generators of the Hall-Janko group as a subgroup of G,(4), J. Algebra 13 (1969), 513-516.

25. J. WALTER, The characterization of finite groups with abelian Sylow 2-subgroups, Ann. Math. 89 (1969), 405-514.

26. H. WIELANDT, “Finite Permutation Groups,” Academic Press, New York/London, 1964.

The irreducible characters of the simple group of M. Suzuki of order 448, 345, 497, 600

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Transcript of The irreducible characters of the simple group of M. Suzuki of order 448, 345, 497, 600