ATLAS: Linear group L_{2}(31)
Order = 14880 = 2^{5}.3.5.31.
Mult = 2.
Out = 2.
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 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 >.
The representations of L_{2}(31) available are:

Permutations on 32 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Some matrix representations in characteristic 2:

Dimension 15 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 32 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 32 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 32 over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 32 over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 32 over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 32 over GF(16):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 31 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 31 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 3 over GF(31):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 31 over GF(31):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some matrix representations in characteristic 0:

Dimension 31 over Z:
a and b (GAP).

Dimension 32 over Z(z15) (monomial):
a and b (GAP).
The representations of 2.L_{2}(31) = SL_{2}(31) available are

Dimension 2 over GF(31):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 16 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 Some matrix representations in characteristic 0:

Dimension 32 over Z(z30) (monomial):
A and B (GAP).
The representations of L_{2}(31):2 available are:

Dimension 30 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
Maximal subgroups
The maximal subgroups of L2(31) are as follows.
 31:15, with generators
???.
 A_{5}
 A_{5}
 D_{32}
 D_{30}
 S_{4}
 S_{4}
The maximal subgroups of L2(31):2 are as follows.
 L_{2}(31)
 31:30
 D_{64}
 D_{60}
A set of generators for the maximal cyclic subgroups of L2(31) can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
A set of generators for the maximal cyclic subgroups of L2(31):2 can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
Go to old L2(31) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 file created on 17th January 2002,
from Version 1 file last modified on 28.10.98.
Last updated 27.06.06 by JNB.
Information checked to
Level 0 on 17.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.