ATLAS: Linear group L_{3}(4), Mathieu group M_{21}, L_{3}(4) subpage
Order = 20160 = 2^{6}.3^{2}.5.7.
Mult = 3 × 4 × 4.
Out = 2 × S_{3} = D_{12}.
The following information is available for covers of L_{3}(4) =
M_{21}:

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.
A presentation of L_{3}(4) on its standard generators is given below. (This one has as its relations powers of short words; it is not so good from
a coset enumeration point of view.)
< a, b  a^{2} = b^{4} =
(ab)^{7} = (ab^{2})^{5} =
(abab^{2})^{7} =
(ababab^{2}ab^{1})^{5} = 1 >.
L3(4) on a and b.
NB: Irreducible, but not absolutely irreducible representations that contain
absolutely irreducible constituents in 2 cohorts are ordered by their
constituents in the `principal' cohort.
The representations of L_{3}(4) available are:
 All primitive permutation representations.

Permutations on 21a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 21b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 56a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 56b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 56c points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 120a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 120b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 120c points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 280 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 2.

Dimension 8 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 8 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 16 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 reducible over GF(4).

Dimension 64 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 3.

Dimension 15a over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15b over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15c over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 19 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45a over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45b over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 63a over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 63b over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 90 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 reducible over GF(9).

Dimension 126 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 reducible over GF(9).
 All faithful irreducibles in characteristic 5.

Dimension 20 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35a over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35b over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35c over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45a over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45b over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 63 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 90 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 reducible over GF(25).
 All faithful irreducibles in characteristic 7.

Dimension 19 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35a over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35b over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35c over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 63a over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 63b over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 126 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 reducible over GF(49).
 Faithful irreducibles in characteristic 0.
The representations of 2.L_{3}(4) available are:
 Some faithful permutation representations.

Permutations on 112a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 112b points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 112c points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 240a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 240b points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 240c points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in characteristic 3.

Dimension 6 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 10a over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 10b over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 20 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(9).

Dimension 22a over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 22b over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 36 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 44 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(9).

Dimension 90 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in characteristic 5.

Dimension 10a over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 10b over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 20 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(25).

Dimension 28 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 36 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 70 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 90 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in characteristic 7.

Dimension 10 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 26 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 28a over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 28b over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(49).

Dimension 64 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 70 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of 3.L_{3}(4) available are:
 Some faithful permutation representations.

Permutations on 63a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 63b points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 360a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 360b points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 360c points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the z3cohort in characteristic 2.

Dimension 3a over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 3b over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 9 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24a over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24b over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles over GF(2).

Dimension 6a over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(4).

Dimension 6b over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(4).

Dimension 18 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(4).

Dimension 48a over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(4).

Dimension 48b over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(4).
 Faithful irreducibles in the z3cohort in characteristic 5.

Dimension 21 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 Faithful irreducibles over GF(5).

Dimension 42 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(25).
 All faithful irreducibles in the z3cohort in characteristic 7.

Dimension 15a over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 15b over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 15c over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 21 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 63a over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 63b over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 84 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 126 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(49).
The representations of 4_{1}.L_{3}(4) available are:
 Some faithful permutation representations.

Permutations on 224 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 480 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of 4_{2}.L_{3}(4) available are:
 Some faithful permutation representations.

Permutations on 224 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 480 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the icohort in characteristic 3.

Dimension 4 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 16 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 28a over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 28b over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 36 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 60 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles over GF(3).

Dimension 8 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(9).

Dimension 32 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(9).

Dimension 56a over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(9).

Dimension 56b over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(9).

Dimension 72 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(9).

Dimension 120 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(9).
 All faithful irreducibles in the icohort in characteristic 5.

Dimension 20 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 28 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 36 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 80a over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 80b over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 160 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(25)
 All faithful irreducibles in the icohort in characteristic 7.

Dimension 20 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 28a over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 28b over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 36 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 44 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles over GF(7).

Dimension 40 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(49).

Dimension 56a over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(49).

Dimension 56b over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(49).

Dimension 72 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(49).

Dimension 88 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(49).
The representations of 6.L_{3}(4) available are:
 Some faithful permutation representations.

Permutations on 720a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 720b points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 720c points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 6 over Z[w]:
A and B (Magma).
The representations of 12_{1}.L_{3}(4) available are:

Permutations on 1440 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of 12_{2}.L_{3}(4) available are:

Permutations on 1440 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the (z3, i)cohort in characteristic 7.

Dimension 12 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 36 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 48 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 84 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The 9 classes of maximal subgroups of L_{3}(4) = M_{21} are as follows:

2^{4}:A_{5} =
M_{20},
with standard generators
b, ab^2ab^3abab^2abab.

2^{4}:A_{5} =
M_{20},
with standard generators
b, abab^2abab^3ab^2ab.

A_{6},
with standard generators a, abab^3abab^2.

A_{6},
with standard generators a, babab^3ab.

A_{6},
with standard generators a, bab^3abab.

L_{3}(2),
with generators a, bab^3ab^3ab.

L_{3}(2),
with generators a, ab^2ab^2abab.

L_{3}(2),
with generators a, ababab^2ab^2.

3^{2}:Q_{8} = M_{9} = U_{3}(2),
with generators
b, ababab^3ab^2abab.
By coincidence, the W1generators for each of the maximal L3(2)s produce them
on their (2, 7, 4)generators (which are unique up to automorphisms for L3(2))
and standard generators for L3(2) may be obtained as (x, xyy) where
(x, y) are the (2, 7, 4)generators. The conversions from W1 to
standard generators have been made available
here,
here and
here.
Representatives of the 10 conjugacy classes of L_{3}(4) = M_{21} are as follows:
 1A: identity [or a^{2}].
 2A: a.^{ }
 3A: abab^{2}ab^{3}abab^{3}.
 4A: b.^{ }
 4B: abab^{3}ab^{2}.
 4C: abab^{2}ab^{3}.
 5A: ab^{2}.
 5B: ab^{2}ab^{2}.
 7A: ab.^{ }
 7B: ab^{3}.
The following information is required and/or useful for the covering groups.
 The top central element of order 3 in 3.L_{3}(4),
6.L_{3}(4), 12_{1}.L_{3}(4),
12_{2}.L_{3}(4), . . . , (3 × 4^{2}).L_{3}(4) is (AB)^{14} = (AB)^{7}.
 The top central element of order 4 in 4_{2}.L_{3}(4)
[and 12_{2}.L_{3}(4)] is (ABABAB^{2}AB^{3})^{15}.
[The above two are determined (from the ABC) by our choices of classes 5A and 7A.]
 We also take the top central element of order 4 in
4_{1}.L_{3}(4) [and 12_{1}.L_{3}(4)] to be (ABABAB^{2}AB^{3})^{15}.
 In 4_{2}.L_{3}(4), B is in class i.4A.
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Version 2.0 created (for main L3(4) page) on 19th September 2001.
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R.A.Wilson, R.A.Parker and J.N.Bray.