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

Order = 20160 = 26.32.5.7.
Mult = 3 × 4 × 4.
Out = 2 × S3 = D12.
WARNING: WORK IN PROGRESS!!!! THIS GROUP HAS LOTS OF PROBLEMS, SO BE PATIENT WHILE WE SORT THEM OUT.

The following information is available on this page for L3(4) = M21:


Index to subpages

This group has so many covers and automorphisms that keeping everything on one page is unwieldy. Therefore we are splitting the pages as shown below.

Standard generators (for the groups L3(4).A only)

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

< a, b | a2 = b4 = (ab)7 = (ab2)5 = (abab2)7 = (ababab2ab-1)5 = 1 >.

< 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


Main ATLAS page Go to main ATLAS (version 2.0) page.
Linear groups page Go to linear groups page.
Old L3(4) page Go to old L3(4) page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 19th September 2001.
Last updated 12.02.02 by JNB.
Information checked to Level 0 on 19.09.01 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.