ATLAS: Unitary group U3(7)

Order = 5663616 = 27.3.73.43.
Mult = 1.
Out = 2.

The following information is available for U3(7):

Standard generators

Standard generators of U3(7) are a and b where a has order 2, b has order 3, ab has order 43 and ababb has order 4.

Type I standard generators of U3(7):2 are c and d where c is in class 7A, d is in class 4D and cd has order 6.
Type II standard generators of U3(7):2 are e and f where e is in class 2B, f has order 3, ef has order 8 and efeff has order 8.

THESE STANDARD GENERATORS (for U3(7):2) ARE STILL PROVISIONAL AND COULD STILL BE CHANGED.

Automorphisms

An outer automorphism of U3(7) can be taken to map (a, b) to (a, bb).

Black box algorithms

To find standard generators for U3(7):
• Find any element of order 6, 12, 24 or 48 [i.e. order divisible by 6]. It powers up to x of order 2 and y of order 3.
[The probability of success at each attempt is 5 in 16 (about 1 in 3).]
• Find a conjugate a of x and a conjugate b of y such that ab has order 43 and ababb has order 4.
[The probability of success at each attempt is 96 in 2107 (about 1 in 22).]
• Now a and b are standard generators for U3(7).
To find standard generators for U3(7).2:
• ... etc.

Presentations

Presentations of U3(7) and U3(7):2 (respectively) on their standard generators are given below.

< a, b | a2 = b3 = (ab)43 = [a, b]4 = [a, bababab]3 = ab(ababab-1)2((ab)3(ab-1)3)3 = 1 >.

< c, d | c7 = d4 = (cd)6 = ... = 1 >.

< e, f | e2 = f3 = (ef)8 = [e, f]8 = (efefef-1)8 = ... = 1 >.

These presentations are available in Magma format as follows: U3(7) on a and b, U3(7):2 on c and d and U3(7):2 on e and f.

Representations

The representations of U3(7) available are:
• Permutation representations, including all primitive ones.
• Permutations on 344 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 688 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1032 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1376 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 2064 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 2107 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 14749 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 16856 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 43904 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Faithful irreducibles in characteristic 2.
• Faithful irreducibles in characteristic 3.
• Faithful irreducibles in characteristic 7.
• Dimension 6 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(49).
• Dimension 8 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 12 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(49).
• Dimension 20 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(49).
• Dimension 27 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 30a over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(49).
• Dimension 30b over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(49).
• Dimension 37 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 42 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(49).
• Dimension 48 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(49).
• Faithful irreducibles in characteristic 43.
• Dimension 43 over Z - NOT YET AVAILABLE: a and b (Magma).
The representations of U3(7):2 available are (NONE):

Maximal subgroups

The maximal subgroups of U3(7) are:
The maximal subgroups of U3(7):2 are [NOT YET FOUND]:
• U3(7), with generators xxx.
• 71+2:48:72 = 71+2:(3 × SD32), with generators xxx.
• 2.(L2(7) × 4).2.2, with generators xxx.
• 82:D12, with generators xxx.
• L2(7):2 × 2, with generators xxx.
• 43:6 = F258, with generators xxx.

Conjugacy classes

The class representatives of the 48 conjugacy classes of Th are as follows:
• 1A: identity or a2.
• 2A: a.
• 3A: b.
• 3B: (ababab2)6 or [a, bab]3.
• 3C: [a, babab]2.
• 4A: ababab2ababab2abab2.
• 4B: (abababab2ab2)3.
• 5A: [a, b]2 or (abab2)2.
• 6A: [a, babab] or (ab)3(ab2)3.
• 6B: (ab)6(ab2)3.
• 6C: (ababab2)3.
• 7A: (ab)6(ab2)6 or (abababab2)3.
• 8A: (ab)7ab2 or ((ab)3ab2ab(ab2)2)3.
• 8B: ababababab2(abab2ab2ab2)2 or (ab)9(ab2)2ab(ab2)3 or ((ab)7ab2ab(ab2)2)3.
• 9A: abab(abab2)3ab2ab2.
• 9B: ababababab2ab2abab2ab2 or ((ab)5ab2)3.
• 9C: (ababab2)2 or [a, bab].
• 10A: [a, b] or abab2.
• 12A/B: abababab2ababab2ab2.
• 12C: (abababab2ab2)2ab2.
• 12D: abababab2ab2.
• 13A: (ab)9(ab2)3 or ab(abababab2)2.
• 14A: (ab)4(ab2)3.
• 15A/B: (ab)6ab2abab2ab2.
• 18A: (ab)10(ab2)4 or abababab2abab2ab2abab2 or (ab)5(ababab2)2.
• 18B: ababab2.
• 19A: ab.
• 20A: (ab)4ab2.
• 21A: abababab2.
• 24A/B: (ab)3ab2ab(ab2)2.
• 24C/D: (ab)7ab2ab(ab2)2.
• 27A: (ab)5ab2.
• 27B/C: (ab)7ab2abab(ab2)3 or ab(abababab2)2abab2.
• 28A: (ab)6ab2.
• 30A/B: (ab)5ab2abab2ab2.
• 31A/B: (ab)5(ab2)2ab(ab2)4 or ab(ababab2)3ab2.
• 36A: (ab)8(ab2)2.
• 36B/C: ab(abababab2)2ab2ab2.
• 39A/B: (ab)8ab2ab(ab2)2.
An element cannot be obtained as a power of an element of greater order just if it is in class 18B or has order at least 19.

A set of generators for the maximal cyclic subgroups can be obtained by running this program [NOT THERE YET] on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy are not dealt with.
Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old U3(7) page - ATLAS version 1.
Anonymous ftp access is also available. See here for details.

Version 2.0 created on 13th October 2003.
Last updated 15.10.03 by JNB.
Information checked to Level 0 on 13.10.03 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.