ATLAS: Unitary group U3(11)

Order = 70915680 = 25.32.5.113.37.
Mult = 3.
Out = S3.

The following information is available for U3(11):

Standard generators

Standard generators of U3(11) are a and b where a has order 2, b has order 3, ab has order 37 and ababb has order 4.
Standard generators of the triple cover 3.U3(11) are preimages A and B where A has order 2 and AB has order 37.

Standard generators of U3(11):2 are c and d where c has order 2 (necessarily class 2B), d has order 4 (necessarily class 4D), cd has order 37 and cdd has order 10.
Standard generators of 3.U3(11):2 are preimages C and D where CD has order 37.

Standard generators of U3(11):3 are not defined.

Standard generators of U3(11):S3 are not defined.

Black box algorithms

To find standard generators for U3(11):
• Find any element of even order. This powers up to x of order 2.
[The probability of success at each attempt is 19 in 32 (about 1 in 2).]
• Find any element of order divisible by 3. This powers up to y of order 3.
[The probability of success at each attempt is 1 in 9.]
• Find conjugates a of x and b of y such that ab has order 37 and ababb has order 4.
[The probability of success at each attempt is 288 in 4477 (about 1 in 16).]
• Now a and b are standard generators for U3(11).

Representations

The representations of U3(11) available are:

The representations of 3.U3(11) available are:

The representations of U3(11):2 available are:

The representations of 3.U3(11):2 available are:

Maximal subgroups

The maximal subgroups of U3(11) are:
The maximal subgroups of U3(11):2 are:
• U3(11).
• 111+2:(5 × 8:2).
• (2.(L2(11) × 2).2).2.
• PGL2(11) × 2.
• PGL2(9).
• (42 × 3):D12.
• 37:6 = F222.
• 32:SD16.
The maximal subgroups of U3(11):3 are:
• U3(11).
• 111+2:120.
• 3 × 2.(L2(11) × 2).2.
• 122:S3.
• 111:3 = F111 × 3.
• 32:2A4.
The maximal subgroups of U3(11):S3 are:
• U3(11):3.
• U3(11):2.
• 111+2:(5 × 24:2).
• (3 × 2.(L2(11) × 2).2).2.
• 122:D12.
• 111:6 = F222 × 3.
• 32:2S4.

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Version 2.0 created on 24th February 2001.
Last updated 17.12.01 by RAW.
Information checked to Level 0 on 24.02.01 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.