ATLAS: Alternating group A_{11}
Order = 19958400 = 2^{7}.3^{4}.5^{2}.7.11.
Mult = 2.
Out = 2.
Standard generators
Standard generators of A_{11} are a
and b where
a is in class 3A, b has order 9
and ab has order 11.
In the natural representation we may take
a = (1, 2, 3) and
b = (3, 4, 5, 6, 7, 8, 9, 10, 11).
Standard generators of the double cover 2.A_{11} are preimages
A
and B where
A has order 3 and B has order 9.
Standard generators of S_{11} = A_{11}.2 are c
and d where
c is in class 2C,
d is in class 10D
and cd has order 11.
In the natural representation we may take
c = (1, 2)
and
d = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11).
Standard generators of either of the double covers 2.S_{11} are preimages
C and D where
CD has order 11.
Representations
The representations of A_{11} available are
- All primitive permutation representations.
- a and
b as
the above permutations on 11 points - the natural representation.
- a and
b as
permutations on 55 points.
- a and
b as
permutations on 165 points.
- a and
b as
permutations on 330 points.
- a and
b as
permutations on 462 points.
- a and
b as
permutations on 2520 points.
- a and
b as
permutations on 2520 points.
- Faithful irreducibles over fields of characteristic 2.
- a and
b as
10 × 10 matrices over GF(2).
- a and
b as
16 × 16 matrices over GF(4).
- a and
b as
16 × 16 matrices over GF(4) - the dual of the above.
- a and
b as
32 × 32 matrices over GF(2) - reducible over GF(4).
- a and
b as
44 × 44 matrices over GF(2).
- a and
b as
100 × 100 matrices over GF(2).
- a and
b as
144 × 144 matrices over GF(2).
- a and
b as
164 × 164 matrices over GF(2).
- a and
b as
186 × 186 matrices over GF(2).
- a and
b as
198 × 198 matrices over GF(2).
- Faithful irreducibles in characteristic 3.
- a and
b as
10 × 10 matrices over GF(3).
- a and
b as
34 × 34 matrices over GF(3).
- a and
b as
45 × 45 matrices over GF(3).
- a and
b as
109 × 109 matrices over GF(3).
- a and
b as
120 × 120 matrices over GF(3).
- a and
b as
126 × 126 matrices over GF(3).
- a and
b as
126 × 126 matrices over GF(3) - the dual of the above.
- a and
b as
131 × 131 matrices over GF(3).
- Faithful irreducibles in characteristic 5.
- a and
b as
10 × 10 matrices over GF(5).
- a and
b as
43 × 43 matrices over GF(5).
- a and
b as
45 × 45 matrices over GF(5).
- a and
b as
55 × 55 matrices over GF(5).
- a and
b as
89 × 89 matrices over GF(5).
- a and
b as
110 × 110 matrices over GF(5).
- a and
b as
120 × 120 matrices over GF(5).
- Faithful irreducibles in characteristic 7.
- a and
b as
10 × 10 matrices over GF(7).
- a and
b as
44 × 44 matrices over GF(7).
- a and
b as
45 × 45 matrices over GF(7).
- a and
b as
66 × 66 matrices over GF(7).
- Faithful irreducibles in characteristic 11.
- a and
b as
9 × 9 matrices over GF(11).
- a and
b as
36 × 36 matrices over GF(11).
- a and
b as
44 × 44 matrices over GF(11).
- a and
b as
84 × 84 matrices over GF(11).
The representations of 2.A_{11} available are
- A and
B as
permutations on 5040 points.
- Faithful irreducibles in characteristic 3.
- A and
B as
16 × 16 matrices over GF(3).
- A and
B as
16 × 16 matrices over GF(3) - the dual of the above.
- A and
B as
144 × 144 matrices over GF(3).
- Faithful irreducibles in characteristic 5.
- A and
B as
16 × 16 matrices over GF(5).
- A and
B as
16 × 16 matrices over GF(5) - the dual of the above.
- A and
B as
56 × 56 matrices over GF(5).
- Faithful irreducibles over fields of characteristic 7.
- A and
B as
16 × 16 matrices over GF(49).
- A and
B as
16 × 16 matrices over GF(49) - the dual of the above.
- A and
B as
32 × 32 matrices over GF(7) - reducible over GF(49).
- A and
B as
144 × 144 matrices over GF(7).
- Faithful irreducibles in characteristic 11.
- A and
B as
16 × 16 matrices over GF(11).
- A and
B as
128 × 128 matrices over GF(11).
The representations of S_{11} = A_{11}:2 available are
- c and
d as
permutations on 11 points - the natural representation.
- There is a primitive permutation representation of S_{11} of degree 362880 on the cosets of the novelty 11:10. However, we decided that this was just a bit too large to keep on file, so if you want this representation, you can make it yourself.
- c and
d as
32 × 32 matrices over GF(2).
The representations of 2.S_{11} (plus type) available are
- C and
D as
32 × 32 matrices over GF(3).
- C and
D as
32 × 32 matrices over GF(5).
- C and
D as
32 × 32 matrices over GF(7).
- C and
D as
16 × 16 matrices over GF(11).
The representations of 2.S_{11} (minus type) available are
Maximal subgroups
The maximal subgroups of A_{11} are:
Return to main ATLAS page.
Last updated 25.08.98
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk