ATLAS: Weyl group Weyl(F4) = GO4+(3)

Order = 1152 = 27.32.
Mult = 22.
Out = D8 [I think].
This group is soluble and has exactly 12 normal subgroups. These subgroups have orders 1, 2 (Z(G) = G''' = Soc(G)), 32 (G'' = O2(G)), 96, 96, 192, 192, 288 (G'), 576, 576, 576 and 1152 (G) respectively.

Standard generators

It seems quite difficult to define standard generators for this group in a sensible manner, but here goes.

Type A standard generators of W(F4) are involutions a, b, c and d such that ab, ac, ad, bc, bd and cd have orders 3, 2, 2, 4, 2 and 3 respectively. These generators satisfy the standard presentation of W(F4).

Type B standard generators of W(F4) are x and y where x is in class 2C (see below), y has order 6, xy has order 6 and xyy has order 4.

We may take x = bd and y = acd.
Conversely, we have a = y3, b = x(xy3)3, c = yxy3xy3x and d = (xy3)3.


Presentations

< a, b, c, d | a2 = b2 = c2 = d2 = (ab)3 = [a, c] = [a, d] = (bc)4 = [b, d] = (cd)3 = 1 >.
The centre is generated by (abcd)6.

< x, y | x2 = y6 = (xy)6 = (xy2)4 = (xyxyxy-2)2 = 1 >.
The centre is generated by [x, y]3.


Representations

The representations of W(F4) available are

Conjugacy classes

There are 25 conjugacy classes of W(F4). The classes are given names as follows: 1A, 2A, 2B, 3A, 3B, 3C, 4A, 6A, 6B, 6C, 12A; 2C, 4B, 4C, 8A; 2D, 2E, 4D, 6D, 6E; 2F, 2G, 4E, 6F, 6G. The semi-colons separate the cosets of G'. I'll give more information in due course.
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Last updated 24th April 1998,
R.A.Wilson, R.A.Parker and J.N.Bray