# ATLAS: Exceptional group G2(4)

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

### Standard generators

Standard generators of G2(4) are a and b where a is in class 2A, b is in class 5C/D, and ab has order 13.
Standard generators of the double cover 2G2(4) are pre-images 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 double cover 2.G2(4):2 are pre-images C and D where CD has order 13.

### Automorphisms

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

### Representations

The representations of G2(4) available are
• a and b as 6 x 6 matrices over GF(4) - the natural representation.
• Some representations in characteristic 3.
• a and b as 64 x 64 matrices over GF(3).
• a and b as 78 x 78 matrices over GF(3).
• a and b as 286 x 286 matrices over GF(3).
• Some representations in characteristic 5.
• a and b as 65 x 65 matrices over GF(5).
• a and b as 78 x 78 matrices over GF(5).
• a and b as 350 x 350 matrices over GF(5).
• a and b as 363 x 363 matrices over GF(5).
• a and b as 650 x 650 matrices over GF(5).
• a and b as 65 x 65 matrices over GF(7).
• a and b as 78 x 78 matrices over GF(7).
• a and b as 65 x 65 matrices over GF(13).
• a and b as 78 x 78 matrices over GF(13).
• Some primitive permutation representations.
• a and b as permutations on 416 points.
• a and b as permutations on 1365 points - the first one in the Atlas.
• a and b as permutations on 1365 points - the second one in the Atlas.
• a and b as permutations on 2016 points.
• a and b as permutations on 2080 points.
• a and b as permutations on 20800 points.
The representations of G2(4):2 available are
• Some representations in characteristic 2.
• c and d as 12 x 12 matrices over GF(2).
• c and d as 28 x 28 matrices over GF(2).
• c and d as 36 x 36 matrices over GF(2).
• c and d as 128 x 128 matrices over GF(2).
• c and d as 168 x 168 matrices over GF(2).
• c and d as 196 x 196 matrices over GF(2).
• c and d as 64 x 64 matrices over GF(3).
• c and d as 65 x 65 matrices over GF(5).
• c and d as 65 x 65 matrices over GF(7).
• c and d as 65 x 65 matrices over GF(13).
The representations of 2.G2(4) available are
• A and B as 12 x 12 matrices over GF(3).
• A and B as 12 x 12 matrices over GF(5).
• A and B as 92 x 92 matrices over GF(5).
• A and B as 12 x 12 matrices over GF(7).
• A and B as 12 x 12 matrices over GF(13).
The representations of 2.G2(4):2 available are
• C and D as 12 x 12 matrices over GF(3) - isoclinic to the Atlas group.
• C and D as 12 x 12 matrices over GF(25) - the Atlas group.
• C and D as 12 x 12 matrices over GF(7).
• C and D as 12 x 12 matrices over GF(169).

### Maximal subgroups

The maximal subgroups of G2(4) are
• J2, with standard generators here.
• 2^2+8:(A5 x 3), with generators here (mapping to standard generators of A5).
• 2^4+6:(A5 x 3)
• U3(4):2, with standard generators here.
• 3.L3(4):2, with (nonstandard) generators here.
• U3(3):2, with (nonstandard) generators here.
• A5 x A5
• L2(13), with standard generators here.
The maximal subgroups of G2(4):2 are