# ATLAS: Linear group L3(5)

Order = 372000 = 25.3.53.31.
Mult = 1.
Out = 2.
The page for the group 53.L3(5) (non-split extension) is available here.

The following information is available for L3(5):

### Standard generators

Type I standard generators of L3(5) are a and b where a has order 3, b is in class 5A and ab has order 20.
Type II standard generators of L3(5) are x and y where x has order 2, y has order 3, xy has order 31 and xyxyy has order 5.

Standard generators of L3(5):2 are c and d where c is in class 2B, d is in class 4D and cd has order 12.

We may obtain .. as x = ((ab)10)babb, y = a, and a' = a = y, b' = ((xyyxyyxyxy)4)yxyyx. The composition of these two maps (either way round) is equivalent to conjugating the generators by an outer element, o say [which is the same either way round], in class 6B, where o-2 = a = y.

### Presentations

Presentations for L3(5) and L3(5):2 on their standard generators are given below.

< a, b | a3 = b5 = aba-1baba-1b2ab-2a-1b2 = abab-2(a-1b2a-1b-2)3 = 1 >.

< x, y | x2 = y3 = (xy)31 = [x, y]5 = ((xy)5(xy-1)4)2 = 1 >.

< c, d | c2 = d4 = (cd)12 = (cdcd2cd2)3 = [d2, cdc]3 = [c, dcdcd-1cdcdcd-1cdcd] = 1 >.

These presentations are available in Magma format as follows: L3(5) on a and b, L3(5) on x and y and L3(5):2 on c and d.

### Representations

The representations of L3(5) available are:
• Some primitive permutation representations.
• Permutations on 31 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 31 points - automorph of the above: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 5.
• Dimension 3 over GF(5) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 3 over GF(5) - the dual of the above: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 8 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
The representations of L3(5):2 available are:
• Permutation representations, including all primitive ones.
• Permutations on 62 points - imprimitive: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 186 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 775 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 3100 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 3875 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 4000 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 6 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 8 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).

### Maximal subgroups

The maximal subgroups of L3(5) are as follows.
• 52:GL2(5).
• 52:GL2(5).
• S5.
• 42:S3.
• F93 = 31:3.
The maximal subgroups of L3(5):2 are as follows. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L3(5) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 28th July 1999.
Last updated 04.02.02 by JNB.
Information checked to Level 0 on 30.07.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.