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

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

### 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 sextuple 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 the double cover 2.U6(2):2 are pre-images C and D where CD has order 11.
• Standard generators of the triple cover 3.U6(2):2 are pre-images C and D where CD has order 11.
• Standard generators of the sixfold cover 6.U6(2):2 are pre-images C and D where CD has order 11.
• Standard generators of 22.U6(2):2 are pre-images 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 pre-images E and F where F has order 11.
• Standard generators of 22.U6(2):3 are pre-images 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', and gh has order 21.
• Standard generators of 3.U6(2):S3 are pre-images G and H. No extra conditions are required, as all such pairs are automorphic.
• Standard generators of 22.U6(2):S3 are pre-images G and H where

### Automorphisms

An automorphism of order 3 can be obtained by mapping (a,b) to ((abb)^-4a(abb)^4, (abababbab)^-1babababbab).
An automorphism of order 2 can be obtained by mapping (a,b) to (a, b^-1).

### Representations

#### U6(2) and covers

• The representations of U6(2) available are
• a and b as permutations on 672 points.
• a and b as permutations on 693 points.
• a and b as permutations on 891 points.
• a and b as permutations on 1408 points.
• a and b as permutations on 1408 points.
• a and b as permutations on 1408 points.
• a and b as permutations on 6237 points.
• a and b as permutations on 20736 points.
• a and b as permutations on 59136 points.
• a and b as 20 × 20 matrices over GF(2).
• a and b as 21 × 21 matrices over GF(3).
• a and b as 22 × 22 matrices over GF(5).
• The representations of 2.U6(2) available are
• A and B as permutations on 1344 points.
• A and B as 56 × 56 matrices over GF(3).
• A and B as 56 × 56 matrices over GF(7).
• The representations of 3.U6(2) available are
• A and B as 6 × 6 matrices over GF(4) - the natural representation.
• A and B as 21 × 21 matrices over GF(7).
• A and B as permutations on 2016 points.
• The representations of 6.U6(2) available are
• A and B as 120 × 120 matrices over GF(7).

#### U6(2):2 and covers

• The representations of U6(2):2 available are
• c and d as 20 × 20 matrices over GF(2).
• c and d as permutations on 1408 points.
• The representation of 2.U6(2):2 available is
• C and D as 56 × 56 matrices over GF(3).
• The representation of 3.U6(2):2 available is
• C and D as 12 × 12 matrices over GF(2).
• The representation of 6.U6(2):2 available is
• C and D as 240 × 240 matrices over GF(7).
• The representations of 22.U6(2):2 available are
• C and D as 112 × 112 matrices over GF(3).
• C and D as 240 × 240 matrices over GF(3).

#### U6(2):3 and covers

• The representation of U6(2):3 available is
• e and f as 20 × 20 matrices over GF(2).
• The representation of 3.U6(2):3 available is
• E and F as 6 × 6 matrices over GF(4).
• The representation of 22.U6(2):3 available is
• E and F as 168 × 168 matrices over GF(3).
• The representation of (22 × 3).U6(2):3 available is
• E and F as 360 × 360 matrices over GF(7).

#### U6(2):S3 and covers

• The representation of U6(2):S3 available is
• g and h as 20 × 20 matrices over GF(2).
• The representation of 3.U6(2).S3 available is
• G and H as 12 × 12 matrices over GF(2).
• The representation of 22.U6(2):S3 available is
• G and H as 168 × 168 matrices over GF(3).
• The representation of (22 × 3).U6(2):S3 available is
• G and H as 720 × 720 matrices over GF(7).

### Maximal subgroups

The maximal subgroups of U6(2) include: