ATLAS: Unitary group U6(2), Fischer group Fi21

Order = 9196830720 = 215.36.5.7.11.
Mult = 22 × 3.
Out = S3.
The information on this page was prepared with help from Ibrahim Suleiman.

The following information is available for U6(2) = Fi21:

[Not linked to yet: this page is still being prepared.]

Standard generators

U6(2) and covers
Standard generators of U6(2) are a and b where a is in class 2A, b has order 7, ab has order 11 and abb has order 18.
Standard generators of the double cover 2.U6(2) are preimages A and B where B has order 7, AB has order 11 and ABBB has order 11.
Standard generators of the triple cover 3.U6(2) are preimages A and B where A has order 2 and B has order 7.
Standard generators of the sixfold cover 6.U6(2) are preimages A and B where A has order 2, B has order 7, AB has order 33 and ABBB has order 11.
Standard generators of 22.U6(2) are preimages A and B where B has order 7 and AB has order 11.
Standard generators of (22 × 3).U6(2) are preimages A and B where A has order 2, B has order 7 and AB has order 33.

U6(2):2 and covers
Standard generators of U6(2):2 are c and d where c is in class 2D, d is in class 6J and cd has order 11.
Standard generators of either double cover 2.U6(2).2 are preimages C and D where CD has order 11.
Standard generators of the triple cover 3.U6(2):2 are preimages C and D where CD has order 11.
Standard generators of either sixfold cover 6.U6(2).2 are preimages C and D where CD has order 11.
Standard generators of 22.U6(2):2 are preimages C and D where C has order 2, D has order 6 and CDCDCDCDCDDCDCDDCDDCDD has order 7.

U6(2):3 and covers
Standard generators of U6(2):3 are e and f where e is in class 3D, f has order 11, ef has order 21 and eff has order 18.
Standard generators of 3.U6(2):3 are preimages E and F where F has order 11.
Standard generators of 22.U6(2):3 are preimages E and F where F has order 11.

U6(2):S3 and covers
Standard generators of U6(2):S3 are g and h where g is in class 2D, h is in class 6J [6J'/6J'' from the point of view of U6(2)] and gh has order 21.
Standard generators of 3.U6(2):S3 are preimages G and H. No extra conditions are required, as all such pairs are automorphic.
Standard generators of 22.U6(2):S3 are preimages G and H where ...

Automorphisms

An automorphism of U6(2) of order 3 can be obtained by mapping (a, b) to ((abb)^-4a(abb)^4, (abababbab)^-1babababbab).
An automorphism of U6(2) of order 2 can be obtained by mapping (a, b) to (a, b^-1).
This automorphism normalises the double cover defined by the standard generators, but interchanges the other two double covers.

Presentations

< a, b | a2 = b7 = (ab)11 = [a, b]2 = [a, b2]3 = [a, b3]3 = (ab3)11 = (abab2ab3ab-3)7 = 1 >.
The last two relations are just quotienting out central involutions from a group of shape 22.U6(2).

Representations

U6(2) and covers

The representations of U6(2) available are: The representations of 2.U6(2) available are: The representations of 3.U6(2) available are: The representations of 6.U6(2) available are: The representations of 22.U6(2) available are: The representations of (22 × 3).U6(2) available are:

U6(2):2 and covers

U6(2):3 and covers

U6(2):S3 and covers


Maximal subgroups

The maximal subgroups of U6(2) are as follows [implementation of word programs not checked]: The maximal subgroups of U6(2):2 are as follows [implementation of word programs not checked]: The maximal subgroups of U6(2):3 are as follows [implementation of word programs not checked]: The maximal subgroups of U6(2):S3 are as follows [implementation of word programs not checked]:

Conjugacy classes

The top central element of order 3 in 3.U6(2) is (AB)11. We can also use (AB)11 as the top central element of order 3 in the covers 6.U6(2) and (22 × 3).U6(2). The element AB is in U6(2)-class 11A.


Main ATLAS page Go to main ATLAS (version 2.0) page.
Classical groups page Go to classical groups page.
Old U6(2) page Go to old U6(2) page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 21st September 2001.
Last updated 03.03.04 by SJN.
Information checked to Level 0 on 21.09.01 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.