# ATLAS: Unitary group U6(3)

Order = 22837472432087040 = 213.315.5.72.13.61.
Mult = 2.
Out = 22.

The following information is available for U6(3):

### Standard generators

Standard generators of U6(3) are a and b where a is in class 2A, b has order 15, ab has order 91 and abb has order 14.
Standard generators of the double cover 2.U6(3) = SU6(3) are preimages A and B where B has order 15 and AB has order 91.

### Black box algorithms

To find standard generators of U6(3):
• Find an element x in class 2A, say by powering up an element of order 122.
• Find an element t of order 15, say by powering up an element of order 30, 60 or 120.
• Look for a conjugate y of t such that xy has order 91 and xyy has order 14.
• The elements x and y are standard generators.

### Presentation

A presentation of U6(3) on its standard generators is given below:

< a, b | a2 = b15 = (ab)91 = (ab2)14 = [a, b]2 = [a, b2]3 = [a, b4]3 = [a, b−2ab5ab2] = [a, b3]4 = 1 >.

This presentation is available in Magma format as follows: U6(3) on a and b.

### Representations

The representations of U6(3) available are:
• Some primitive permutation representations.
• Some irreducibles in characteristic 3.
• Dimension 15a over GF(9): A and B (Magma). — Λ2(6a).
• Dimension 15b over GF(9): A and B (Magma). — Λ2(6b).
• Dimension 21a over GF(9): A and B (Magma). — S2(6a).
• Dimension 21b over GF(9): A and B (Magma). — S2(6b).
• Dimension 30 over GF(3): A and B (Magma). — reducible over GF(9).
• Dimension 34 over GF(3): A and B (Magma). — the adjoint representation.
• Dimension 42 over GF(3): A and B (Magma). — reducible over GF(9).
The representations of 2.U6(3) available are:
• Some irreducibles in characteristic 3.
• Dimension 6a over GF(9): A and B (Magma). — the natural module.
• Dimension 6b over GF(9): A and B (Magma). — the Frobenius automorph of the above.
• Dimension 12 over GF(3): A and B (Magma). — reducible over GF(9).
• Dimension 20 over GF(3): A and B (Magma). — exterior cube of a natural module.
• Dimension 50a over GF(9): A and B (Magma). — in 6a × 15a.
• Dimension 50b over GF(9): A and B (Magma). — the Frobenius automorph of the above.
• Dimension 100 over GF(3): A and B (Magma). — reducible over GF(9).

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Version 2.0 file created on 4th August 2004.
Last updated 27.05.08 by JNB.