# ATLAS: Linear group L2(31)

Order = 14880 = 25.3.5.31.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of L2(31) are a and b where a has order 2, b has order 3 and ab has order 31.
Standard generators of the double cover 2.L2(31) = SL2(31) are preimages A and B where B has order 3 and AB has order 31.

Standard generators of L2(31):2 = PGL2(31) are c and d where c has order 2, d has order 3, cd has order 10 and cdcdd has order 8.
Standard generators of either of the double covers 2.L2(31).2 = 2.PGL2(31) are preimages C and D where D has order 3.

### Presentations

Presentations for L2(31) and L2(31):2 = PGL2(31) in terms of their standard generators are given below.

< a, b | a2 = b3 = [a,b]3ab[a,b][a,bab][a,b-1abab] = 1 >.

< c, d | c2 = d3 = (cd)10 = [c, d]8 = (cd(cdcdcd-1)3)2 = 1 >.

### Representations

The representations of L2(31) available are:
• a and b as permutations on 32 points.
• a and b as 15 × 15 matrices over GF(2).
• a and b as 3 × 3 matrices over GF(31).
The representations of 2.L2(31) = SL2(31) available are
• A and B as 2 × 2 matrices over GF(31).
• A and B as 16 × 16 matrices over GF(5).