# ATLAS: Alternating group A14

Order = 43589145600 = 210.35.52.72.11.13.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of A14 are a and b where a is in class 3A, b has order 13, ab has order 12, abb has order 24 and ababb has order 20. The last two conditions may be replaced by [a, b] has order 2.
In the natural representation we may take a = (1, 2, 3) and b = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14).
Standard generators of the double cover 2.A14 are preimages A and B where A has order 3 and B has order 13.

Standard generators of S14 = A14:2 are c and d where c is in class 2D, d has order 13 and ab has order 14.
In the natural representation we may take c = (1, 2) and d = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14).
Standard generators of either of the double covers 2.S14 are preimages C and D where D has order 13.

In the natural representations given here, we have a = cd-1cd = [c, d] and b = d.

### Automorphisms

An outer automorphism of A14 may be realised by mapping (a, b) to (a-1, ba-1). In the natural representations given here, this outer automorphism is conjugation by c.

### Representations

The representations of A14 available are:
• a and b as the above permutations on 14 points.
• a and b as permutations on 91 points.
• a and b as permutations on 364 points.
The representations of 2.A14 available are:
• A and B as 32 × 32 matrices over GF(49).
• A and B as 32 × 32 matrices over GF(49) - the automorph of the above.
• A and B as 64 × 64 matrices over GF(7) - reducible over GF(49).
The representations of S14 = A14:2 available are:
• c and d as the above permutations on 14 points.
• c and d as permutations on 91 points.
• c and d as permutations on 364 points.
• c and d as 64 × 64 matrices over GF(2) - a constituent of the permutation representation on 91 points.
• c and d as 64 × 64 matrices over GF(2) - the spin representation.
The representations of 2.S14 (plus type) available are:
• C and D as 64 × 64 matrices over GF(7).
The representations of 2.S14 (minus type) available are:
• none.

### Maximal subgroups

The maximal subgroups of A14 are as follows:
• A13.
Order: 3113510400.
Index: 14.

• S12 = A12:2.
Order: 479001600.
Index: 91.

• (A11 × 3):2.
Order: 119750400.
Index: 364.

• (A10 × A4):2.
Order: 43545600.
Index: 1001.

• (A7 × A7):4.
Order: 25401600.
Index: 1716.

• (A9 × A5):2.
Order: 21772800.
Index: 2002.

• (A8 × A6):21.
Order: 14515200.
Index: 3003.

• 26:S7.
Order: 322560.
Index: 135135.

• L2(13).
Order: 1092.
Index: 39916800.

The maximal subgroups of S14 are as follows:
• S13.
Order: 6227020800.
Index: 14.

• S12 × 2.
Order: 958003200.
Index: 91.

• S11 × S3.
Order: 239500800.
Index: 364.

• S10 × S4.
Order: 87091200.
Index: 1001.

• (S7 × S7):2 = S7 wr 2.
Order: 50803200.
Index: 1716.

• S9 × S5.
Order: 43545600.
Index: 2002.

• S8 × S6.
Order: 29030400.
Index: 3003.

• 27:S7 = 2 wr S7.
Order: 645120.
Index: 135135.

• PGL2(13) = L2(13):2.
Order: 2184.
Index: 39916800.

### Conjugacy classes

The 72 conjuagcy classes of A14 are as follows:
Class Cycle type p power p' part Order of centraliser Centraliser Representative
1A 114 - - 43589145600 A14 identity
2A 110.22 A A 14515200 (A10 × 22):2 [a, b]
2B 16.24 A A 138240
2C 12.26 A A 46080
3A 111.3 A A 59875200 A11 × 3 a
3B 18.32 A A 362880 (A8 × 32):2
3C 15.33 A A 9720
3D 12.34 A A 1944
4A 18.2.4 A A 161280
4B 16.42 B A 11520
4C 25.4 A A 7680
4D 14.23.4 A A 2304
4E 2.43 C A 384
4F 12.22.42 B A 256
5A 19.5 A A 907200 A9 × 5
5B 14.52 A A 600 (A4 × 52):2
21A 14.3.7 AA AA 252 A4 × 21
21B 1.32.7 AB AB 63 32 × 7
24A 1.2.3.8 24 24
24B 6.8 24 24 ab2 or ab-5
28A 1.2.4.7 AA AA 28 28
30A 12.22.3.5 120 15 × D8
30B 1.2.5.6 30 30
33A 3.11 AA AA 33 33
33B* 3.11 AA AA 33 33 ab4 or ab-6
35A 12.5.7 AA AA 35 35
42A 22.3.7 84 21 × 22
45A 5.9 45 45 ab5 or ab-3
45B* 5.9 45 45
60A 2.3.4.5 60 60

The 135 conjuagcy classes of S14 are as follows:
Class Cycle type Order of centraliser Centraliser Representative
2A110.22
2B16.24
2C12.26