ATLAS: Exceptional group F4(2)
Order = 3311126603366400.
Mult = 2.
Out = 2.
Standard generators
Standard generators of F4(2) are
a
and b where
a is in class 2C,
b is in class 3C,
ab has order 17,
and ababababbababbabb has order 13.
An image of (a,b) under the outer automorphism
is given by
((ab)^5a(ab)^5, (abb)^3bb(abb)^3).
Standard generators of the double cover 2.F4(2) are preimages
A
and B where
B has order 3
and AB has order 17.
Standard generators of F4(2):2 are
c
and d where
c is in class 2E,
d is in class 3AB,
cd has order 40 (in fact this is in class 40B),
and abababb has order 10.
Standard generators of the double cover 2.F4(2):2 are preimages
C
and D where
D has order 3.
Representations
The representations of F4(2) available are

a and
b as
26 x 26 matrices over GF(2).

a and
b as
permutations on 69888 points.
The representation of F4(2).2 available is

c and
d as
52 x 52 matrices over GF(2).
The representations of 2.F4(2) available are

A and
B as
52 x 52 matrices over GF(3).

A and
B as
52 x 52 matrices over GF(5).

A and
B as
permutations on 139776 points.

A and
B as
2380 x 2380 matrices over GF(3).
The representations of 2.F4(2).2 available are

C and
D as
52 x 52 matrices over GF(25).

C and
D as
52 x 52 matrices over GF(5). These matrices actually generate an isoclinic group
2.(F4(2) x 2).2, and NOT 2.F4(2).2.
Maximal subgroups
Maximal subgroups of F4(2) include:
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Last updated 29.10.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk