ATLAS: Unitary group U_{6}(3)
Order = 22837472432087040 = 2^{13}.3^{15}.5.7^{2}.13.61.
Mult = 2.
Out = 2^{2}.
The following information is available for U_{6}(3):
Standard generators of U_{6}(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.U_{6}(3) = SU_{6}(3)
are preimages A and B where
B has order 15 and AB has order 91.
To find standard generators of U_{6}(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.
A presentation of U_{6}(3) on its standard generators is given below:
< a, b  a^{2} = b^{15} = (ab)^{91} =
(ab^{2})^{14} = [a, b]^{2} =
[a, b^{2}]^{3} = [a, b^{4}]^{3} =
[a, b^{−2}ab^{5}ab^{2}] =
[a, b^{3}]^{4} = 1 >.
This presentation is available in Magma format as follows:
U_{6}(3) on a and b.
The representations of U_{6}(3) available are:

Some primitive permutation representations.

Permutations on 22204 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
A and B (Magma).

Permutations on 44226 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
A and B (Magma).

Some irreducibles in characteristic 3.

Dimension 15a over GF(9):
A and B (Magma).
— Λ^{2}(6a).

Dimension 15b over GF(9):
A and B (Magma).
— Λ^{2}(6b).

Dimension 21a over GF(9):
A and B (Magma).
— S^{2}(6a).

Dimension 21b over GF(9):
A and B (Magma).
— S^{2}(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.U_{6}(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.