ATLAS: Linear group L5(2)

Order = 9999360 = 210.32.5.7.31.
Mult = 1.
Out = 2.

Dempwolff group 25.L5(2) (non-split extension)

Order = 319979520 = 215.32.5.7.31.
Mult = 1.
Out = 1.

Standard generators

Standard generators of L5(2) are a and b where a is in class 2A, b has order 5 and ab has order 21.

Standard generators of the Dempwolff group 25.L5(2) are preimages A and B such that A has order 2, B has order 5, ABABB has order 10, ABABABBBB has order 28 and ABABABBBABB has order 5.
(NB: All involutions of 25.L5(2) not in the normal 25 map onto class 2A of L5(2).)

Standard generators of L5(2):2 are c and d where c is in class 2C, d is in class 8B, cd has order 21 and cdcdd has order 8. (NB: it is not possible to have d in 8C, with these other properties.)


Automorphisms

An outer automorphism of L5(2) of order 2 may be obtained by mapping (a, b) to (a, b-1).

Standard generators of L5(2):2 may be obtained as follows: c is the given automorphism and d = cab.
To return to L5(2), the pair (a', b') = (dddd, (cdd)^-1cdcdcddcdcdcddcdcdd) is equivalent to (a, b) (i.e. they are conjugate in L5(2):2).


Presentations

Presentations of L5(2), 25.L5(2) and L5(2):2 on their standard generators are given below.

< a, b | a2 = b5 = (ab)21 = [a, b]4 = [a, b2]2 = (ababab-2)4 = 1 >.

< A, B | A2 = B5 = (AB)21 = [A, B2]4 = [A, B-2AB-2][A, B]3 = (ABABAB-2)2(ABAB-1AB)2(AB-1)2 = 1 >.

< c, d | c2 = d8 = (cd)21 = (cd4)4 = [c, d]5 = [c, d2]2 = (cdcdcdcd-2)2 = 1 >.


Representations

The representations of L5(2) available are: The representations of 25.L5(2) available are: The representations of L5(2):2 available are:

Maximal subgroups

The maximal subgroups of L5(2) are as follows. The maximal subgroups of L5(2):2 are as follows.
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Last updated 25th March 1999,
R.A.Wilson, R.A.Parker and J.N.Bray