# ATLAS: Exceptional group ^3D4(2)

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

### 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.
• a and b as 8 x 8 matrices over GF(8) - the natural representation.
• a and b as 26 x 26 matrices over GF(2).
• Some representations in characteristic 3.
• a and b as 25 x 25 matrices over GF(3).
• a and b as 52 x 52 matrices over GF(3).
• Some representations in characteristic 7.
• a and b as 26 x 26 matrices over GF(7).
• a and b as 298 x 298 matrices over GF(7).
• Some representations in characteristic 13.
• a and b as 26 x 26 matrices over GF(13).
• Some permutation representations.
• a and b as permutations on 819 points.
The representations of ^3D4(2):3 available are
• c and d as 24 x 24 matrices over GF(2).
• c and d as 26 x 26 matrices over GF(2).
• c and d as 144 x 144 matrices over GF(2).
• c and d as 52 x 52 matrices over GF(3).
• c and d as 196 x 196 matrices over GF(3).

### Maximal subgroups

The maximal subgroups of ^3D4(2) are
The maximal subgroups of ^3D4(2):3 are
• ^3D4(2)
• 2^1+8:L2(8):3
• [2^11]:(7:3 x S3)
• 3 x U3(3):2
• S3 x L2(8):3
• (7:3 x L2(7)):2
• 3^1+2.2S4.3
• 7^2:(2A4 x 3)
• 3^2:2A4 x 3
• 13:12

Return to main ATLAS page. Last updated 24.03.99

R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk