# 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

• Standard generators of L3(4) are a and b where a has order 2, b has order 4, ab has order 7 and abb has order 5.
• Standard generators of the triple cover 3.L3(4) are preimages A and B where A has order 2 and B has order 4.
• Standard generators of the double cover 2.L3(4) are preimages A and B where AB has order 7, ABB has order 5 and ABABABBB has order 5.
• Standard generators of the quadruple cover 4a.L3(4) are preimages A and B where B has order 4, AB has order 7 and ABB has order 5.
• Standard generators of the quadruple cover 4b.L3(4) are preimages A and B where B has order 4, AB has order 7 and ABB has order 5.
• Standard generators of the sextuple cover 6.L3(4) are preimages A and B where AB has order 2, B has order 4, AB has order 21, ABB has order 5 and ABABABBB has order 5.
• Standard generators of the twelvefold cover 12a.L3(4) are preimages A and B where A has order 2, B has order 4, AB has order 21 and ABB has order 5.
• Standard generators of the twelvefold cover 12b.L3(4) are preimages A and B where A has order 2, B has order 4, AB has order 21 and ABB has order 5.

#### L3(4):2a and covers

• Standard generators of L3(4):2a are c and d where c is in class 2B, d is in class 4D and cd has order 7.
• Standard generators of the triple cover 3.L3(4):2a are preimages C and D where C has order 2, and D has order 4.
• Standard generators of the quadruple cover 4a.L3(4):2a are preimages C and D where CD has order 7 ... and some other condition(s).

#### L3(4):2b and covers

• Standard generators of L3(4):2b are e and f where e is in class 2C, f has order 5 and ef has order 14, and eff has order 8.

#### L3(4):6 and covers

• Standard generators of L3(4):6 are k and l where k is in class 2B, l is in class 6C, kl has order 21, and klkll has order 4.

#### L3(4):S3a and covers

• Standard generators of L3(4):S3a = L3(4):3:2b are o and p where o is in class 2C, p is in class 3C and op has order 14.
• Standard generators of any of the triple covers 3.L3(4).S3a = L3(4).3:2b are preimages O and P. No further conditions are needed.

#### L3(4):S3b and covers

• Standard generators of L3(4):S3b = L3(4):3:2c are q and r where q is in class 2D, r is in class 3C, qr has order 8 and qrqrr has order 15.
• Standard generators of any of the triple covers 3.L3(4).S3b = L3(4).3:2c are preimages Q and R. No further conditions are needed.

#### L3(4):D12 and covers

• Standard generators of L3(4):D12 are s and t where s is in class 2D, t is in class 4E, st has order 6, stt has order 6 and ststtsttt has order 8.
• Standard generators of any of the triple covers 3.L3(4).D12 are preimages S and T. No other conditions are needed.

### 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.
• The representations of L3(4) available are
• a and b as 9 × 9 matrices over GF(2).
• a and b as 8 × 8 matrices over GF(4).
• a and b as 15 × 15 matrices over GF(3).
• a and b as 19 × 19 matrices over GF(7).
• a and b as permutations on 21 points.
• The representations of 3.L3(4) available are
• A and B as 3 × 3 matrices over GF(4) - the natural representation.
• A and B as permutations on 63 points.
• The representations of 4_1.L3(4) available are
• A and B as 8 × 8 matrices over GF(9).
• A and B as 8 × 8 matrices over GF(5).
• A and B as 8 × 8 matrices over GF(49).
• A and B as permutations on 480 points.
• The representations of 4_2.L3(4) available are
• A and B as 4 × 4 matrices over GF(9).
• The representations of 6.L3(4) available are
• A and B as 6 × 6 matrices over GF(7).
• A and B as 6 × 6 matrices over GF(25).
The representations of 12_1.L3(4) available are
• A and B as 24 × 24 matrices over GF(25).
• A and B as 24 × 24 matrices over GF(49).
The representations of 12_2.L3(4) available are
• A and B as 36 × 36 matrices over GF(25).
• A and B as 12 × 12 matrices over GF(49).

#### L3(4):2a and covers

• The representation of L3(4):2a available is
• c and d as 18 × 18 matrices over GF(2).
The representation of 3.L3(4):2a available is
• C and D as 6 × 6 matrices over GF(4).
The representation of 4a.L3(4):2a available is
• C and D as 16 × 16 matrices over GF(5).

#### L3(4):2b and covers

• The representation of L3(4):2b available is
• e and f as 9 × 9 matrices over GF(2).

#### L3(4):2c and covers

• The representation of 4_1.L3(4):2_3 available is
• g and h as 16 × 16 matrices over GF(7).

#### L3(4):6 and covers

• The representations of L3(4):6 available are
• k and l as 19 × 19 matrices over GF(3).
• k and l as 45 × 45 matrices over GF(3).
• The representations of groups isoclinic to 4^2.L3(4).6 available are
• K and L as 24 × 24 matrices over GF(9).
• K and L as 48 × 48 matrices over GF(49).

#### L3(4):D12 and covers

• The representations of L3(4):D12 available are
• s and t as permutations on 42 points.
• s and t as 18 × 18 matrices over GF(2) - changed to standard generators on 24/10/98.
• s and t as 19 × 19 matrices over GF(3) - changed to standard generators on 24/10/98. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L3(4) page - ATLAS version 1. 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.