ATLAS: Exceptional group ^{3}D_{4}(3)
Order = 20560831566912 = 2^{6}.3^{12}.7^{2}.13^{2}.73.
Mult = 1.
Out = 3.
The following information is available for ^{3}D_{4}(3):
Standard generators of ^{3}D_{4}(3) are a and b
where a is in class 3A, b is in class 13G/H (the ones with fixed points
in the natural 8dimensional representation), ab has order 73
and ababb has order 13.
(These conditions distinguish between classes 13G and 13H.)
Add the condition that abb has order 73 if you can't distinguish between 13G/H
and 13I/J/K. Note that (3A, 13A/B/C/D/E/F, 73)triples are impossible.
Standard generators of ^{3}D_{4}(2):3 are
not yet defined.
Note that elements of orders 42, 78, 84 power up to 3A. Nothing properly powers up
to 13G/H or 13I/J/K. These classes contain 5/169 elements of the group,
and 2/169 of the group is in class 13G/H. 13A/B/C/D/E/F has centraliser 13 × L_{3}(3),
so obtaining such elements without powering up is extremely unlikely.
The representations of ^{3}D_{4}(3) available are:

Dimension 8 over GF(27)  the natural representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 218 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 26572 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The maximal subgroups of ^{3}D_{4}(3) are:
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 10th October 2000.
Last updated 11.11.08 by JNB.
Information checked to
Level 0 on 10.10.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.