ATLAS: Linear group L₃(4), Mathieu group M₂₁, L₃(4).2₁ subpage

Order = 40320 = 2⁷.3².5.7.
Mult = 3 × 4 × 4.
Out = S₃.

The following information is available for covers of L₃(4).2₁:

Standard generators.
Presentations.
Representations.
Maximal subgroups.
Conjugacy classes.

Standard generators

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.

Presentations

Representations

The representations of L₃(4).2₁ 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.
- None yet.

The representations of 2.L₃(4).2₁ 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.L₃(4).2₁ available are:

Some faithful permutation representations.
All faithful irreducibles in the z3-cohort 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 z3-cohort 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 z3-cohort 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 4₁.L₃(4).2₁ available are:

Some faithful permutation representations.
All faithful irreducibles in the i-cohort 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 i-cohort 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 i-cohort 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 4₂.L₃(4).2₁ available are:

Some faithful permutation representations.
All faithful irreducibles in the i-cohort 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 i-cohort 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 i-cohort 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.L₃(4).2₁ available are:

Some faithful permutation representations.
All faithful irreducibles in the z3-cohort 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 z3-cohort 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 12₁.L₃(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 12₂.L₃(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).

Maximal subgroups

Conjugacy classes

Representatives of the conjugacy classes of L₃(4).2₁ are as follows:

1A: CC
2A: DD
3A: 6A²
4A: CDDCDD
4B: 8B²
4C: 8C²
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.

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