ATLAS: Exceptional group G₂(4)

Order = 251596800 = 2¹².3³.5².7.13.
Mult = 2.
Out = 2.

See also ATLAS of Finite Groups, pp97-99.

Standard generators

Standard generators of G₂(4) are a and b where a is in class 2A, b is in class 5C/D, ab has order 13 and abb has order 13.
To ensure that b is in class 5C/D (and not class 5A/B), also check that ababb has order 15.
The representations below are labelled such that b is in class 5C of the Atlas of Brauer Characters.
Standard generators of the double cover 2.G₂(4) are preimages A and B where B has order 5 and AB has order 13.

Standard generators of G₂(4):2 are c and d where c is in class 2C, d is in class 4D, cd has order 13 and cdcddd has order 6.
Standard generators of the Atlas double cover 2.G₂(4):2 are preimages C and D where CD has order 13.
(Similarly for the isoclinic, nonAtlas, double cover 2.G₂(4).2.)

Automorphisms

The outer automorphism of G₂(4) can be realised by the map taking (a, b) to (a, b²).

Black box algorithms

To find standard generators of G₂(4):

Find an element x of order 2 by taking a suitable power of any element of order 4, 8 or 12.
[The probability of success at each attempt is 33 in 256 (about 1 in 8).]
Find an element y of order 5 by taking a suitable power of any element of order 5, 10 or 15.
[The probability of success at each attempt is 12 in 25 (about 1 in 2), and the porbability that y is in the right class is 1 in 2.]
Find conjugates a of x and b of y such that ab has order 13 and abb has order 13.
[The probability of success at each attempt is 20 in 273 (about 1 in 14).]
If ababb has order 15, then y was in the right class and a and b are standard generators of G₂(4). Otherwise ababb has order 10 and y was in the wrong class, so go back to Step 2.

To find standard generators of G₂(4).2:

Find any element of order 14. Its seventh power, x say, is in class 2C.
[The probability of success at each attempt is 1 in 14 (or 1 in 7 if we search through outer elements only).]
Er, I don't know how to find a 4D-element. 15/64 of the outer elements have order 4 or 12, and among these just 1/5 power into class 4D.
Suppose we have obtained y in class 4D.
Find conjugates c of x and d of y such that cd has order 13 and cdcddd has order 6.
[The probability of success at each attempt is 6 in 325 (about 1 in 54).]
Now c and d are standard generators of G₂(4).2.

Alternatively, to find standard generators of G₂(4).2:

Find any element of order 14. Its seventh power, x say, is in class 2C.
[The probability of success at each attempt is 1 in 14 (or 1 in 7 if we search through outer elements only).]
Find an element y of order 13.
[The probability of success at each attempt is 1 in 13 (or 2 in 13 if we search through inner elements only).]
Find conjugates c of x and z of y such that cz has order 4, czz has order 6, czzz has order 6.
[The probability of success at each attempt is 1 in 1600.]
Now c and d = cz are standard generators of G₂(4).2.

Representations

The representations of G₂(4) available are:

Some representations in characteristic 2:
- Dimension 6 over GF(4) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 6 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 14 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 14 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 36 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 64 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 64 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 84 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 84 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 196 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 384 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 896 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Some representations in characteristic 3.
- Dimension 64 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 78 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 286 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Some representations in characteristic 5.
- Dimension 65 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 78 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 350 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 363 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Dimension 650 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 65 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 78 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 65 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 78 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Some faithful irreducibles in characteristic 0
- Dimension 65 over Z: a and b (GAP).
- Dimension 78 over Z: a and b (GAP).
- Dimension 350 over Z: a and b (GAP).
Some primitive permutation representations.
- Permutations on 416 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Permutations on 1365[a] points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Permutations on 1365[b] points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Permutations on 2016 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Permutations on 2080 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- Permutations on 20800 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

The representations of 2.G₂(4) available are:

Dimension 12 over GF(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 12 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 92 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 12 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Dimension 12 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
Some faithful irreducibles in characteristic 0
- Dimension 12 over Z[i]: A and B (GAP).

The representations of G₂(4):2 available are:

Some representations in characteristic 2.
- Dimension 12 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
- Dimension 28 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
- Dimension 36 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
- Dimension 128 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
- Dimension 168 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
- Dimension 196 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 64 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 65 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 65 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
Dimension 65 over GF(13): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).

The representations of 2.G₂(4):2 available are:

Dimension 12 over GF(9): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
Dimension 12 over GF(25): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
Dimension 12 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
Dimension 12 over GF(169): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).

The representations of the nonsplit 2.G₂(4).2 available are:

Dimension 12 over GF(3): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).

Maximal subgroups

The maximal subgroups of G₂(4) are:

J₂, with standard generators here.
2²⁺⁸:(A₅ × 3), with generators here (mapping to standard generators of A₅).
2⁴⁺⁶:(A₅ × 3), with generators here.
U₃(4):2, with standard generators here.
3.L₃(4):2, with (nonstandard) generators here.
U₃(3):2, with (nonstandard) generators here.
A₅ × A₅, with generators here.
L₂(13), with standard generators here.

The maximal subgroups of G₂(4):2 are:

G₂(4), with standard generators here.
J₂:2.
2²⁺⁸:(A₅ × 3):2.
2⁴⁺⁶:(A₅ × 3):2.
U₃(4):4.
3.L₃(4).2.2.
U₃(3):2 × 2.
(A₅ × A₅):2.
L₂(13):2.

Conjugacy classes

A set of generators for the maximal cyclic subgroups of G₂(4) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

A set of generators for the maximal cyclic subgroups of G₂(4):2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

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Version 2.0 created on 14th April 2000.
Last updated 02.03.11 by JNB.
Information checked to Level 0 on 14.04.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.