ATLAS: Linear group L_{2}(31)
Order = 14880 = 2^{5}.3.5.31.
Mult = 2.
Out = 2.
Standard generators
Standard generators of L_{2}(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.L_{2}(31) =
SL_{2}(31) are preimages A and B where B has
order 3 and AB has order 31.
Standard generators of L_{2}(31):2 = PGL_{2}(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.L_{2}(31).2 =
2.PGL_{2}(31) are preimages C and D where D has
order 3.
Presentations
Presentations for L_{2}(31) and L_{2}(31):2 = PGL_{2}(31) in terms of their standard generators are given below.
< a, b | a^{2} = b^{3} = [a,b]^{3}ab[a,b][a,bab][a,b^{-1}abab] = 1 >.
< c, d | c^{2} = d^{3} = (cd)^{10} = [c, d]^{8} = (cd(cdcdcd^{-1})^{3})^{2} = 1 >.
Representations
The representations of L_{2}(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.L_{2}(31) = SL_{2}(31) available are
- A and
B as
2 × 2 matrices over GF(31).
- A and
B as
16 × 16 matrices over GF(5).
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Last updated 28th October 1998,
R.A.Wilson, R.A.Parker and J.N.Bray