ATLAS: Linear group L_{3}(4), Mathieu group M_{21}, L_{3}(4).2_{1} subpage
Order = 40320 = 2^{7}.3^{2}.5.7.
Mult = 3 × 4 × 4.
Out = S_{3}.
The following information is available for covers of L_{3}(4).2_{1}:

Standard generators of L3(4).2a are c and d where c is in class 2B
2, 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 double cover 2.L3(4).2a are preimages
C
and D where
CD has order 7, and CDCDCDDCDCDD has order 3.

Standard generators of 4a.L3(4).2a are preimages
C
and D where
CD has order 7, and CDD has order 8.

Standard generators of 4b.L3(4).2a are preimages
C
and D where
CD has order 7 and CDD has order 8.

Standard generators of the sextuple cover 6.L3(4).2a are preimages
C
and D where
C has order 2, D has order 4, CD has order 21, and CDCDCDDCDCDD has order 3.

Standard generators of 12a.L3(4).2a are preimages
C
and D where
C has order 2, D has order 4, CD has order 21 and CDD has order 8.

Standard generators of 12b.L3(4).2a are preimages
C
and D where
C has order 2, D has order 4, CD has order 21 and CDD has order 8.
The representations of L_{3}(4).2_{1} available are:
 All faithful irreducibles in characteristic 2.

Dimension 16 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 18 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

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

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

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

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

Dimension 19 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 90 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 126 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 5.

Dimension 20 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

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

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

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

Dimension 63 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

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

Dimension 19 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

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

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

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

Dimension 45 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 126 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Faithful irreducibles in characteristic 0.
The representations of 2.L_{3}(4).2_{1} available are:
 Some faithful permutation representations.
 All faithful irreducibles in characteristic 3.

Dimension 6 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 20 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 36 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 44 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 90 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in characteristic 5.

Dimension 20 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 28 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 36 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 70 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 90 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in characteristic 7.

Dimension 10 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 26 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 56 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 64 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 70 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 3.L_{3}(4).2_{1} available are:
 Some faithful permutation representations.
 All faithful irreducibles in the z3cohort in characteristic 2.

Dimension 6 over GF(4):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 9 over GF(4):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 48 over GF(4):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in the z3cohort in characteristic 5.

Dimension 15 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 15 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 15 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 21 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 63 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 90 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in the z3cohort in characteristic 7.

Dimension 15 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 15 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 15 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 21 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 126 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 4_{1}.L_{3}(4).2_{1} available are:
 Some faithful permutation representations.
 All faithful irreducibles in the icohort in characteristic 3.

Dimension 16 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 16 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 48 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 48 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 80 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in the icohort in characteristic 5.

Dimension 16 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 112 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 160 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 160 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in the icohort in characteristic 7.

Dimension 16 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 16 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 112 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 128 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 4_{2}.L_{3}(4).2_{1} available are:
 Some faithful permutation representations.
 All faithful irreducibles in the icohort in characteristic 3.

Dimension 8 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 32 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 56 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 56 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 72 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 120 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in the icohort in characteristic 5.

Dimension 40 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 56 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 72 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 160 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 160 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in the icohort in characteristic 7.


Dimension 40 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 56 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 56 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 72 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 88 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 6.L_{3}(4).2_{1} available are:
 Some faithful permutation representations.
 All faithful irreducibles in the z3cohort in characteristic 5.

Dimension 6 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 36 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 90 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 120 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in the z3cohort in characteristic 7.

Dimension 6 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 36 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 54 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 84 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 12_{1}.L_{3}(4) available are:
 All faithful irreducibles in the (z3, i)cohort in characteristic 5.

Dimension 48 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 48 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 240 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in the (z3, i)cohort in characteristic 7.

Dimension 48 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 96 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 96 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 12_{2}.L_{3}(4) available are:
 All faithful irreducibles in the (z3, i)cohort in characteristic 5.

Dimension 72 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 96 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 120 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 120 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 All faithful irreducibles in the (z3, i)cohort in characteristic 7.

Dimension 24 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 72 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 96 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 168 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
Representatives of the conjugacy classes of L_{3}(4).2_{1} are as follows:
 1A: CC
 2A: DD
 3A: 6A^{2}
 4A: CDDCDD
 4B: 8B^{2}
 4C: 8C^{2}
 5AB: CDCDDCDD
 7AB: CD
 2B: C
 4D: D
 6A: CDCDDCDDCDCDDCDDCDD
 8A: CDD
 8B: CDCDCDDCDDD
 8C: CDCDDCDDCDCDDCDDD
This program makes these representatives.
The following information is required and/or useful for the covering groups.
 In 2.G.2, the above class representatives are +5AB, +7AB, +6A, +8A, 8B, 8C.
 In 3.G.2, the top central element is (CD)^{7}.
 In both groups 4.G.2, the centre has order 2 and the two elements of order 4 are conjugate.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
Go to main L3(4) page.
Go to old L3(4) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created (for main L3(4) page) on 19th September 2001.
Subpage last updated 12.01.05 by RAW.
Information checked to
Level 0 on 19.09.01 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.