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