# ATLAS: Linear group L5(2)

Order = 9999360 = 210.32.5.7.31.
Mult = 1.
Out = 2.

#### Dempwolff group 25.L5(2) (non-split extension)

Order = 319979520 = 215.32.5.7.31.
Mult = 1.
Out = 1.

### Standard generators

Standard generators of L5(2) are a and b where a is in class 2A, b has order 5 and ab has order 21.

Standard generators of the Dempwolff group 25.L5(2) are preimages A and B such that A has order 2, B has order 5, ABABB has order 10, ABABABBBB has order 28 and ABABABBBABB has order 5.
(NB: All involutions of 25.L5(2) not in the normal 25 map onto class 2A of L5(2).)

Standard generators of L5(2):2 are c and d where c is in class 2C, d is in class 8B, cd has order 21 and cdcdd has order 8. (NB: it is not possible to have d in 8C, with these other properties.)

### Automorphisms

An outer automorphism of L5(2) of order 2 may be obtained by mapping (a, b) to (a, b-1).

Standard generators of L5(2):2 may be obtained as follows: c is the given automorphism and d = cab.
To return to L5(2), the pair (a', b') = (dddd, (cdd)^-1cdcdcddcdcdcddcdcdd) is equivalent to (a, b) (i.e. they are conjugate in L5(2):2).

### Presentations

Presentations of L5(2), 25.L5(2) and L5(2):2 on their standard generators are given below.

< a, b | a2 = b5 = (ab)21 = [a, b]4 = [a, b2]2 = (ababab-2)4 = 1 >.

< A, B | A2 = B5 = (AB)21 = [A, B2]4 = [A, B-2AB-2][A, B]3 = (ABABAB-2)2(ABAB-1AB)2(AB-1)2 = 1 >.

< c, d | c2 = d8 = (cd)21 = (cd4)4 = [c, d]5 = [c, d2]2 = (cdcdcdcd-2)2 = 1 >.

### Representations

The representations of L5(2) available are:
• a and b as permutations on 31 points.
• a and b as permutations on 155 points.
• a and b as 5 × 5 matrices over GF(2) - the natural representation.
• a and b as 30 × 30 matrices over GF(3).
• a and b as 124 × 124 matrices over GF(3).
• a and b as 30 × 30 matrices over GF(5).
• a and b as 123 × 123 matrices over GF(5).
• a and b as 30 × 30 matrices over GF(7).
• a and b as 94 × 94 matrices over GF(7).
• a and b as 29 × 29 matrices over GF(31).
• a and b as 124 × 124 matrices over GF(31).
The representations of 25.L5(2) available are:
• A and B as permutations on 7440 points - on the cosets of C(A) = 24.24.L3(2) - CFs of rep are 1+30+30+124+217+280+868+930+1240+3720.
• A and B as permutations on 7440 points.
• A and B as permutations on 7440 points.
• A and B as 248 × 248 matrices over GF(3).
• A and B as 248 × 248 matrices over GF(5).
• A and B as 248 × 248 matrices over GF(7).
• A and B as 248 × 248 matrices over Z.
• This last representation is also available in MeatAxe format as A and B .
The representations of L5(2):2 available are:
• c and d as permutations on 62 points.
• c and d as 10 × 10 matrices over GF(2).
• c and d as 30 × 30 matrices over GF(7).
• c and d as 29 × 29 matrices over GF(31). This representation can be used to distinguish the classes 8B and 8C, which have traces 14 and 15 respectively.
• c and d as 30 × 30 matrices over Z[r2].

### Maximal subgroups

The maximal subgroups of L5(2) are as follows.
The maximal subgroups of L5(2):2 are as follows.