# ATLAS: Exceptional group 3D4(2)

Order = 211341312 = 212.34.72.13.
Mult = 1.
Out = 3.

The following information is available for 3D4(2):

### Standard generators

Standard generators of 3D4(2) are a and b where a is in class 2A, b has order 9, ab has order 13 and abb has order 8.
The last condition may be replaced by: b is in class 9A and ab is in class 13A.
Standard generators of 3D4(2):3 are c and d where c has order 2, d is in class 3D, cd has order 21, cdcdd has order 7 and cdcdcdcddcdcddcddcdd has order 6.
Note: c is in class 2B.
10/2/99: standard generators of 3D4(2):3 corrected. There are no elements of the group satisfying the previous definition.

### Representations

The representations of 3D4(2) available are:
• Some representations in characteristic 2.
• Dimension 8 over GF(8) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 26 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some representations in characteristic 3.
• Some representations in characteristic 7.
• Some representations in characteristic 13.
• Some permutation representations.
The representations of 3D4(2):3 available are:

### Maximal subgroups

The maximal subgroups of 3D4(2) are:
• 21+8:L2(8), with generators here.
• [211]:(7 × S3), with generators here.
• U3(3):2, with generators here.
• S3 × L2(8), with generators here.
• (7 × L2(7)):2, with generators here.
• 31+2.2S4, with generators here.
• 72:2A4, with generators here.
• 32:2A4, with generators here.
• 13:4, with generators here.
The maximal subgroups of 3D4(2):3 are:
• 3D4(2).
• 21+8:L2(8):3.
• [211]:(7:3 × S3).
• 3 × U3(3):2, with generators here.
• S3 × L2(8):3.
• (7:3 × L2(7)):2.
• 31+2.2S4.3.
• 72:(2A4 × 3).
• 32:2A4 × 3.
• 13:12.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of 3D4(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.

A set of generators for the maximal cyclic subgroups of 3D4(2):3 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

An outer automorphism of 3D4(2) can be obtained by running this program on the standard generators.

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