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.


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Version 2.0 created on 21st September 2001.
Last updated 15.04.05 by RAW.