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 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.
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.
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.
The representations of U6(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).
— 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.