# ATLAS: Unitary group U4(3)

Order = 3265920 = 27.36.5.7.
Mult = 32 × 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 2.U4(3) are preimages A and B where ABBB has order 5 and ABABB has order 7.
• Standard generators of the quadruple cover 4.U4(3) = SU4(3) are preimages A and B where ABBB has order 5 and ABABB has order 7.
These generators were chosen so that A and B have orders 2 and 6 respectively; this choice forces AB to have order 28.
• Standard generators of the triple cover 31.U4(3) are preimages A and B where A has order 2, AB has order 7 and ABAB3AB4 has order 5.
• Standard generators of the triple cover 32.U4(3) are preimages A and B where A has order 2, AB has order 7 and ABABBB has order 7.
• Standard generators of the quadruple cover 32.U4(3) are preimages A and B where A has order 2 and AB has order 7.
• Standard generators of (P × Q).U4(3), where P is a 2-group and Q is a 3-group, map onto standard generators of both P.U4(3) and Q.U4(3).

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

#### U4(3).21 and covers

• Standard generators of U4(3).21 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/E', f has order 9, ef has order 28 and efeff has order 6.

#### U4(3).22 and covers

• Standard generators of U4(3).22 are g and h where g is in class 2D, h has order 7 and gh has order 8.

#### U4(3).23 and covers

• Standard generators of U4(3).23 are i and j where i is in class 2F, j has order 5, ij has order 8, ijj has order 8, ijijj has order 7 and ijijjijjj has order 10.
• Standard generators of 32.U4(3).23 are preimages I and J where I has order 2 and J has order 5.
• Standard generators of 32.U4(3).23' are preimages I and J where [I has order 2 and] J has order 5.

#### U4(3).D8 and covers

• Standard generators of U4(3):D8 are o and p where o is in class 2DD', p is in class 6NN' and op has order 20.
We may obtain p as p = ox, where x has order 20 and ox has order 6.
• Standard generators of 32.U4(3):D8 are preimages O and P where ... the representations given below are on them.
• Standard generators of 2.U4(3).D8 (containing 2.U4(3)) are preimages O and P. No further conditions are needed.
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).

#### U4(3).D8 and covers

• The representations of U4(3):D8 available are:
• Permutations on 112 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - primitive - on the cosets of 3^4:(A6:D8).
• Permutations on 252 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - imprimitive - on the cosets of U4(2):2 × 2.
• Permutations on 280 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - primitive.
• Permutations on 324 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - imprimitive - on the cosets of L3(4):2^2.
• Permutations on 540 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - primitive - on the cosets of (U3(3) × 4):2.
• Permutations on 1134 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - imprimitive - on the cosets of 2^5:S6.
• Permutations on 2835 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - primitive.
• Permutations on 4536 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - primitive.
• Permutations on 5184 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - imprimitive - on the cosets of S7.
• Permutations on 8505 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - primitive.
• Permutations on 9072 points: o and p (Meataxe), o and p (Meataxe binary), o and p (GAP). - imprimitive - on the cosets of C(p^3) = A6.2^2 × 2.
• The representations of 32.U4(3):D8 available are:
• Permutations on 756a points: O and P (Meataxe), O and P (Meataxe binary), O and P (GAP). - on the cosets of C(O) = (U4(2) × 3):2 × 2.
• Permutations on 756b points: O and P (Meataxe), O and P (Meataxe binary), O and P (GAP). - on the cosets of U4(2):2 × S3.
• Permutations on 972 points: O and P (Meataxe), O and P (Meataxe binary), O and P (GAP). - on the cosets of 3.L3(4).2^2.
• The representations of 2.U4(3).D8 available are:
The version of 2.U4(3).D8 we have is GO6-(3).2, the subgroup of GL6(3) fixing or negating a non-degenerate orthogonal form of minus type. This group seems not to have a double cover of type 4.U4(3).D8.

### Maximal subgroups Go to main ATLAS (version 2.0) page. Go to classical groups page. Go to old U4(3) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 25th May 2004, from a version 1 file last updated on 29th Octocber 1999.
Last updated 28.06.07 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.