Hiss , Lux and Parker

Then the element (^12:2^)^ has order three and lies in the conjugacy class 3F, which is the preimage of the G-class 3B, as can be checked by using the mutation representation of 3.G and the number of fixed points of this element in the permutation representation. We denote this element shortly by SB. The subgroup generated by ZB and y where y is

is 3 X (7з(3) in 3.G. If we multiply у by a suitable central 3-element z then ZB and X :=:yz generate t/J3(3). Now take the subgroup U of С/з(3) generated by ZB and (3Bx3Bxx)<^^^^^^3^^3^^3^^*>. Then C/ is a group of isomorphism type 3^"*"^: 8. We are going to use U as condensation subgroup.

Let P denote the permutation module of fc[3.G] on the cosets of the subgroup 2As> Let i denote the central idempotent of fc[3.G] given by i = (1 + z + z^)/Z {z is a central element of order 3). Then P ^ Pi^ P{\ — i) and P(l — i) is the faithful component of P. The ordinary character of P(l — i) in the block Bi is 2772 + 5103 + 6ЗЗ61 + 8064. Hence it can be written in terms of our basic set BSi:

126 -h 126# -h 639 + 846 + 1035

+ 126 + 126^^ -f 153 + 639 + 2178 + 846 -f 1035

- hl26 + 126^^ + 153 + 639 + 2178 + 1233 + 846 + 1035

+ 126 + 126^ + 153 + 639 + 1233 + 4752 + 1035.

Here all Brauer characters are known to be irreducible besides 2178 which might split as 945 + 1233 and 4752 which might split as 3501 + 153 + 153 + 945.

Let и = (X^a-gt; ^)I\U\ denote the trace element of the condensation group и. We want to study the ufc[3.G]u-module P(l — i)u. If X is any section of P(l — «), the dimension of Xu is the dimension of the space of fixed points of t/ on X, since the order of (7 is not divisible by 5. This dimension can be culated by using ordinary characters. The degrees, dimensions of (/-fixed spaces and the multiplicities of the Brauer characters in BSi occurring in P(l — i) are as follows:

126 126 153 639 1233 2178 1035 846 4752 1113 5 9 7 5 20

44342 2 431

If 2178 would spHt as 945 + 1233 the fixed dimension of 945 would be 4.

Since we shall work over the field with 5 elements, the degrees of the sponding characters have to be doubled and also the dimensions of the [/-fixed spaces. Thus P(l — %)и has dimension 228. Using the Meat-Axe we get the following irreducible composition factors of P(l — i)u and their multiplicities:

2a 26 2c 6a 10a 106 14a 18a 40a 4344234 2 1

108