ATLAS: Linear group L_{3}(4), Mathieu group M_{21}
Order = 20160.
Mult = 3 × 4 × 4.
Out = 2 × S3.
Standard generators
Note
Standard generators for groups of the form M.L_{3}(4).A, where M.L_{3}(4) is a cover of L_{3}(4) and L_{3}(4).A is a subgroup of Aut(L_{3}(4)) = L_{3}(4):D_{12} will be defined to be preimages of standard generators of L_{3}(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
< o, p  o^{2} = p^{3} = (op)^{14} = [o, p]^{6} = [o, pop]^{4} = ((op)^{6}(op^{1})^{2})^{3} = 1 >.
< s, t  s^{2} = t^{4} = (st)^{6} = (st^{2})^{6} = (stst^{2})^{6} = (st^{2}[s, t]^{3})^{2} = (stst^{2}st^{2})^{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.
Return to main ATLAS page.
Last updated 1st June 2000,
R.A.Wilson, R.A.Parker and J.N.Bray