# ATLAS: Unitary group U7(2)

Order = 227787103272960 = 221.38.5.7.11.43.
Mult = 1.
Out = 2.
The page for the group 214.U7(2) (non­split extension) is available here.

The following information is available for U7(2):

### Standard generators

Standard generators of U7(2) are a and b where a is in class 2A, b has order 7, ab has order 33 and abb has order 45.

NB: Class 2A is the class of transvections of U7(2).

### Automorphisms

An outer automorphism of U7(2) may be obtained by mapping (a, b) to (a, b-1).

### Presentation

A presentation of U7(2) on its standard generators is given below.

< a, b | a2 = b7 = (ab)33 = [a, b]3 = [a, b2]3 = [a, b3]3 = [a, bab]2 = [a, b2ab2]2 = [a, bab2]3 = (abab-3ab-3)8 = 1 >.

This presentation is available in Magma format as follows: U7(2) on a and b.

### Representations

The representations of U7(2) available are:

### Maximal subgroups

The maximal subgroups of U7(2) are as follows [from Kleidman's list]:
• 21+10:(3 × U5(2)), with generators a, bab^3ab.
Order: 84085309440.
Index: 2709.

• 3.U6(2).3, with generators a, bab^5.
Order: 82771476480.
Index: 2752.

• 29+6:3.L3(4).3.
Order: 5945425920.
Index: 38313.

• 24+12.(A5 × 31+2:2A4).
Order: 2548039680.
Index: 89397.

• 3 × S3 × U5(2).
Order: 246343680.
Index: 924672.

• 31+2:2A4 × U4(2).
Order: 16796160.
Index: 13561856.

• 36:S7, with generators a, babab^3abab^4.
Order: 3674160.
Index: 61997056.

• 43:7 = F301.
Order: 301.
Index: 756767784960.

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Version 2.0 created on 8th December 1999.
Last updated 02.08.00 by JNB.
Information checked to Level 0 on 09.12.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.