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 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.
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).
The representations of U3(11) available are:
-
Permutations on 1332 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 110 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 370[a] over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 370[b] over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 370[c] over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 8 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 10 over GF(121):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 3.U3(11) available are:
-
Dimension 3 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of U3(11):2 available are:
-
Permutations on 1332 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 110 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 8 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 3.U3(11):2 available are:
-
Dimension 6 over GF(11):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
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.