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.)
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
Standard generators of the image group PGO4+(3) are
images of standard generators of W(F4), and have been given the
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.
The representations of W(F4) available are
- x and
permutations on 24 points.
- x and y
as 4 × 4 matrices over Z.
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
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.