ATLAS: Exceptional group ^{3}D_{4}(2)
Order = 211341312 = 2^{12}.3^{4}.7^{2}.13.
Mult = 1.
Out = 3.
The following information is available for ^{3}D_{4}(2):
Standard generators of ^{3}D_{4}(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 ^{3}D_{4}(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 ^{3}D_{4}(2):3 corrected. There are no elements
of the group satisfying the previous definition.
The representations of ^{3}D_{4}(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.

Dimension 25 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 52 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 196 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 324 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 351 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 351 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 351 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 441 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 1053 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some representations in characteristic 7.

Dimension 26 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 298 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some representations in characteristic 13.

Dimension 26 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some permutation representations.

Permutations on 819 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of ^{3}D_{4}(2):3 available are:
 Some representations in characteristic 2.

Dimension 24 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 26 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 144 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 246 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 480 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 52 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 196 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 26 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 52 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 273 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 298 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 467 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 26 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The maximal subgroups of ^{3}D_{4}(2) are:
The maximal subgroups of ^{3}D_{4}(2):3 are:

^{3}D_{4}(2).

2^{1+8}:L_{2}(8):3.

[2^{11}]:(7:3 × S_{3}).

3 × U_{3}(3):2, with generators
here.

S_{3} × L_{2}(8):3.

(7:3 × L_{2}(7)):2.

3^{1+2}.2S_{4}.3.

7^{2}:(2A_{4} × 3).

3^{2}:2A_{4} × 3.

13:12.
A set of generators for the maximal cyclic subgroups of
^{3}D_{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.
A set of generators for the maximal cyclic subgroups of
^{3}D_{4}(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
^{3}D_{4}(2)
can be obtained
by running this program on the standard
generators.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old 3D4(2) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 17th April 2000.
Last updated 04.11.02 by RAW.
Information checked to
Level 0 on 18.04.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.