# ATLAS: Group 2F4(2)'

## An Apology

I must apologise to the eminent mathematician whose name is usually attached to this group for removing his name from this page and those linked to or from it. The reason is that certain web-crawlers which have been scanning my pages have misinterpreted the occurrence of this name as an indication of quite a different content on these pages from that which actually pertains.

Sorry.

Order = 17971200 = 211.33.52.13.
Mult = 1.
Out = 2.

### Standard generators

Standard generators of the group 2F4(2)' are a and b where a is in class 2A, b has order 3 and ab has order 13.

Standard generators of its automorphism group 2F4(2) = 2F4(2)'.2 are c and d where c is in class 2A, d is in class 4F, cd has order 12 and cdcddcddd has order 4.

A pair of elements conjugate to (a, b) may be obtained as a' = , b' = .

### Representations

The representations of 2F4(2)' available are:
• All irreducible representations in characteristic 2 (up to automorphisms)
• a and b as 26 × 26 matrices over GF(2).
• a and b as 246 × 246 matrices over GF(2).
• a and b as 2048 × 2048 matrices over GF(4).
• Some representations in characteristic 3
• a and b as 26 × 26 matrices over GF(3).
• a and b as 27 × 27 matrices over GF(9).
• a and b as 77 × 77 matrices over GF(3).
• Some representations in characteristic 5
• a and b as 26 × 26 matrices over GF(25).
• a and b as 27 × 27 matrices over GF(5).
• a and b as 78 × 78 matrices over GF(5).
• a and b as 109 × 109 matrices over GF(25).
• Some representations in characteristic 13
• a and b as 26 × 26 matrices over GF(169).
• a and b as 27 × 27 matrices over GF(13).
• a and b as 52 × 52 matrices over GF(13) - reducible over GF(169).
• a and b as 78 × 78 matrices over GF(13).
• All primitive permutation representations (up to automorphisms)
• a and b as permutations on 1600 points.
• a and b as permutations on 1755 points.
• a and b as permutations on 2304 points.
• a and b as permutations on 2925 points.
• a and b as permutations on 12480 points.
• a and b as permutations on 14976 points.
The representations of 2F4(2) = 2F4(2)'.2 available are as follows.
NB: generators changed to standard generators on 18.08.97!!!
• Some representations in characteristic 2
• c and d as 26 × 26 matrices over GF(2) - the natural representation.
• Some representations in characteristic 3
• c and d as 52 × 52 matrices over GF(3).
• c and d as 54 × 54 matrices over GF(3).
• c and d as 77 × 77 matrices over GF(3).
• Some representations in characteristic 5
• c and d as 27 × 27 matrices over GF(5).
• c and d as 52 × 52 matrices over GF(5).
• c and d as 78 × 78 matrices over GF(5).
• c and d as 218 × 218 matrices over GF(5).
• Some representations in characteristic 13
• c and d as 27 × 27 matrices over GF(13).
• Some permutation representations
• c and d as permutations on 1755 points.
• c and d as permutations on 2304 points.

### Maximal subgroups

The maximal subgroups of ^2F4(2)' are
The maximal subgroups of ^2F4(2) = ^2F4(2)'.2 are
• ^2F4(2)'
• 13:12
• 3^1+2:SD16
• 2.[2^9].5.4
• L2(25).2
• 2^2.[2^9].S3
• 5^2:4S4