# ATLAS: Unitary group U4(3)

Order = 3265920.
Mult = 3 x 3 x 4.
Out = D8.

### Standard generators

#### U4(3) and covers

• Standard generators of U4(3) are a and b where a has order 2, b is in class 6A, ab has order 7, and abababbababb has order 5.
• Standard generators of the double cover 2U4(3) are pre-images A and B where B has order and AB has order .

The following are images of the standard generators under certain outer automorphisms:

#### U4(3).2_1 and covers

• Standard generators of U4(3).2_1 are c and d where c is in class 2B, d has order 9, cd has order 14, and cdcdd has order 9.

#### U4(3).4 and covers

• Standard generators of U4(3).4 are e and f where e is in class 4E, f has order 9, ef has order 28, and efeff has order 6.
The full envelope of U4(3) is 4.3^2.U4(3).D8. The bicyclic extensions are as follows:
together with some others that I forgot, such as 3_2.U4(3).2_3' and the like.

Other extensions for which representations are available are:

• none yet.

### Representations

#### U4(3) and covers

• The representations of U4(3) available are
• a and b as permutations on 112 points.
• a and b as permutations on 162 points.
• a and b as permutations on 567 points.
• a and b as permutations on 1296 points.
• a and b as 20 x 20 matrices over GF(2).
• a and b as 19 x 19 matrices over GF(3).
• a and b as 35 x 35 matrices over GF(5).
• a and b as 90 x 90 matrices over GF(5).
The representations of 2.U4(3) available are
• A and B as 120 x 120 matrices over GF(5).
The representations of 3_1.U4(3) available are
• A and B as 6 x 6 matrices over GF(4).
• A and B as 21 x 21 matrices over GF(25).
The representations of 3_2.U4(3) available are
• A and B as 36 x 36 matrices over GF(4).
• A and B as 36 x 36 matrices over GF(25).
The representations of 3^2.U4(3) available are
• A and B as 12 x 12 matrices over GF(4) - reducible.
The representations of 6_1.U4(3) available are
• A and B as 6 x 6 matrices over GF(25).
The representations of 12_1.U4(3) available are
• A and B as 84 x 84 matrices over GF(25).
• A and B as 132 x 132 matrices over GF(25).
The representations of 12_2.U4(3) available are
• A and B as 36 x 36 matrices over GF(25).
The representations of 4.U4(3) available are
• A and B as 4 x 4 matrices over GF(9) - the natural representation.
• A and B as 120 x 120 matrices over GF(5).

#### U4(3).2_1 and covers

• The representations of U4(3).2_1 available are
• c and d as 20 x 20 matrices over GF(2).
• c and d as 140 x 140 matrices over GF(2).
• The representations of 12_2.U4(3).2_1 available are
• C and D as 72 x 72 matrices over GF(5).

#### U4(3).4 and covers

• The representations of U4(3).4 available are
• e and f as 20 x 20 matrices over GF(2).