# ATLAS: Linear group L2(8)

Order = 504 = 23.32.7.
Mult = 1.
Out = 3.

The following information is available for L2(8):

### Standard generators

Standard generators of L2(8) are a and b where a has order 2, b has order 3 and ab has order 7.

Standard generators of L2(8):3 are c and d where c has order 2, d has order 3, cd has order 9 and cdcdcddcdcddcdd has order 7. The last condition is equivalent to: cdcdcddcddcdcdd has order 2. Note that these conditions imply that d is conjugate to the field automorphism that squares the matrix entries in the natural representation of L2(8).

### Black box algorithms

To find standard generators for L2(8):
• Find any elements x of order 2 and y of order 3.
• Find a conjugate a of x and a conjugate b of y whose product has order 7.
• The elements a and b are now standard generators of L2(8).
To find standard generators for L2(8).3:
• Find any element of order 6. This cubes to x of order 2 and squares to y of order 3.
• Find a conjugate c of x and a conjugate d of y whose product has order 9.
• If cdcdcddcdcddcdd has order 2 then invert d.
• The elements c and d are now standard generators of L2(8):3.

### Automorphisms

An outer automorphism of L2(8) of order 3 may be obtained by mapping (a, b) to (a, bababba).

To obtain our standard generators for L2(8):3 we may take c = babb and d to be the above automorphism.
Conversely, we may take a = cd-1cdcd-1cd-1cdcdcd-1cdc and b = cdcdcd-1cd-1cd-1cdcd-1cd. Note also that a' = c and b' = d-1(cd)3d are equivalent under an automorphism to (a, b).

### Presentations

Presentations for L2(8) and L2(8):3 = R(3) in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)7 = (ababab-1ababab-1ab-1)2 = 1 >.

< c, d | c2 = d3 = (cd)9 = [c, d]9 = (cdcdcd-1cd-1cdcd-1)2 = 1 >.

### Representations

The representations of L2(8) available are
• All primitive permutation representations.
• a and b as permutations on 9 points.
• a and b as permutations on 28 points.
• a and b as permutations on 36 points.
• a and b as 2 × 2 matrices over GF(8) - the natural representation.
The representations of L2(8):3 available are
• All faithful primitive permutation representations.
• c and d as permutations on 9 points.
• c and d as permutations on 28 points.
• c and d as permutations on 36 points.
• All faithful irreducible representations in characteristic 2 whose character appears in the ABC.
• c and d as 6 × 6 matrices over GF(2).
• c and d as 8 × 8 matrices over GF(2).
• c and d as 12 × 12 matrices over GF(2).
• Both faithful irreducible representations in characteristic 3.
• c and d as 7 × 7 matrices over GF(3) - the natural representation as R(3).
• c and d as 27 × 27 matrices over GF(3).
• All faithful irreducible representations in characteristic 7 whose character appears in the ABC.
• c and d as 7 × 7 matrices over GF(7).
• c and d as 8 × 8 matrices over GF(7).
• c and d as 21 × 21 matrices over GF(7).
• All faithful irreducible representations in characteristic 0 whose character appears in the ATLAS.
• c and d as 7 × 7 matrices over Z.
• c and d as 8 × 8 matrices over Z.
• c and d as 21 × 21 matrices over Z.
• c and d as 27 × 27 matrices over Z.

### Maximal subgroups

The maximal subgroups of L2(8) are as follows.
The maximal subgroups of L2(8):3 are as follows.
• L2(8).
• 2^3:7:3.
• 9:6.
• 7:6.

### Conjugacy classes

The following are conjugacy class representatives of L2(8).
• 1A: identity.
• 2A: a.
• 3A: b.
• 7A: ab.
• 7B: abab.
• 7C: ababab or abababab.
• 9A: ababb.
• 9B: ababbababb.
• 9C: abababbabb or (ababb)^4.
The following are conjugacy class representatives of L2(8):3.
• 1A: identity.
• 2A: c.
• 3A: cdcdcd.
• 7ABC: cdcdcddcdd.
• 9ABC: cdcdd.
• 3B: d.
• 6A: cdcdcdd.
• 9D: cd.
• 3B': dd.
• 6A': cdcddcdd.
• 9D': cdd.