ATLAS: Linear group L3(4), Mathieu group M21

Order = 20160.
Mult = 3 × 4 × 4.
Out = 2 × S3.

Standard generators

Note

Standard generators for groups of the form M.L3(4).A, where M.L3(4) is a cover of L3(4) and L3(4).A is a subgroup of Aut(L3(4)) = L3(4):D12 will be defined to be preimages of standard generators of L3(4).A (with some extra conditions). We will get round to defining standard generators for all these groups in due course.

L3(4) and covers

L3(4):2a and covers

L3(4):2b and covers

L3(4):6 and covers

L3(4):S3a and covers

L3(4):S3b and covers

L3(4):D12 and covers


Automorphisms

Automorphisms of L3(4)

An automorphism of type 2_1 can be obtained by mapping (a,b) to (a, (abb)^-1babb).
An automorphism of type 2_2 can be obtained by mapping (a,b) to ((ab)^3b(ab)^3, b).
An automorphism of order 3 can be obtained by mapping (a,b) to (a, (abb)^-2babababbab(abb)^4).
An automorphism of order 6 can be obtained by mapping (a,b) to ((ab)^-2bab, (abb)^-2babababbab(abb)^4) .

Automorphisms of L3(4):2_1

An automorphism of order 3 can be obtained by mapping (c,d) to (d^-1cd, (abb)^-3b(abb)^3).

Presentations

< o, p | o2 = p3 = (op)14 = [o, p]6 = [o, pop]4 = ((op)6(op-1)2)3 = 1 >.

< s, t | s2 = t4 = (st)6 = (st2)6 = (stst2)6 = (st2[s, t]3)2 = (stst2st2)6 = 1 >.


Representations

L3(4) and covers

NB. The representations of the covers of L3(4) have now been adjusted to conform to the definitions of standard generators given above.

L3(4):2a and covers

L3(4):2b and covers

L3(4):2c and covers

L3(4):6 and covers

L3(4):D12 and covers


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Last updated 1st June 2000,
R.A.Wilson, R.A.Parker and J.N.Bray