ATLAS: Weyl group W(F4), Orthogonal group GO4+(3)

Group W(F4) = GO4+(3)
Order = 1152 = 27.32.
Mult = 22.
Out = D8 [I think]. 		Group PGO4+(3) = (A4 × A4):22
Order = 576 = 26.32.
Mult = 22.
Out = 2. 		Group O4+(3) = A4 × A4
Order = 144 = 24.32.
Mult = 22 × 3.
Out = D8.

The following information is available for W(F4):

The group G = W(F4) 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.

We shall give faithful representations of W(F4) = GO4+(3) = 21+4:(S3 × S3) and PGO4+(3) = (A4 × A4):22 = 24:(S3 × S3). Any other image of W(F4) is a quotient of S3 × S3. (However, note that PGO4+(3) is not isomorphic to S4 × S4.)

Standard generators

Type I 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 II 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 = y3xy3xy3, c = yxy3xy3x and d = (xy3)3 = (y3x)3.
(These maps are exact inverses of each other, not merely inverses up to automorphisms.)

Standard generators of the image group PGO4+(3) are images of standard generators of W(F4), and have been given the same labelling.

Presentations

Presentations of W(F4) on its standard generators are given below. Quotienting out by the given central elements gives rise to presentations of PGO4+(3) on its standard generators.

< 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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Version 2.0 created on 14th February 2004, from Version 1 file last modified on 24.04.98.
Last updated 11.04.05 by RAW.
Information checked to Level 0 on 16.02.04 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.