# ATLAS: Alternating group A14

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

The following information is available for A14:

### 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.

### Black box algorithms

To find standard generators for A14:
• Find an element of order 33, 42 or 60. This powers to x in class 3A.
[The probability of success at each attempt is 103 in 1155 (about 1 in 11).]
• Find an element y of order 13.
[The probability of success at each attempt is 2 in 13 (about 1 in 7).]
• Find conjugates a of x and b of y such that ab has order 12 and [a, b] has order 2.
[The probability of success at each attempt is 1 in 56.]
• Now a and b are standard generators of A14.
To find standard generators for S14 = A14.2:
• Find an element of order 22 or 70. This powers to x in class 2D.
[The probability of success at each attempt is 23 in 385 (about 1 in 17) or 46 in 385 (about 1 in 8) if you look through outer elements only.]
• Find an element y of order 13.
[The probability of success at each attempt is 1 in 13 or 2 in 13 (about 1 in 7) if you look through inner elements only.]
• Find conjugates c of x and d of y such that cd has order 14.
[The probability of success at each attempt is 1 in 7.]
• Now c and d are standard generators of S14.

### Presentations

Presentations for A14 and S14 (respectively) on their standard generators are given below.

< a, b | a3 = b13 = (ab)12 = [a, b]2 = (aabab)2 = [a, babab]2 = (aabababab)2 = [a, babababab]2 = 1 >.

< c, d | c2 = d13 = (cd)14 = [c, d]3 = [c, dcd]2 = [c, dcdcd]2 = [c, (cd)4]2 = [c, (cd)5]2 = 1 >.

These presentations, and those of the covering groups, are available in Magma format as follows:
A14 on a and b, 2A14 on A and B, S14 on c and d, 2S14 (+) on C and D and 2S14 (-) on C and D.

### Representations

Representations are available for the following decorations of A14. The representations of A14 available are:
• Primitive permutation representations.
• Permutations on 14 points - the natural representation above: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 91 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 364 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1001 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1716 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 2002 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 3003 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 135135 points: a and b (Magma).
• Faithful irreducibles in characteristic 2.
• Dimension 12 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 64 over GF(2) - a constituent of the permutation representation on 91 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 64 over GF(2) - the spin representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Faithful irreducibles in characteristic 3.
• Faithful irreducibles in characteristic 5.
• Faithful irreducibles in characteristic 7.
• Faithful irreducibles in characteristic 11.
• Faithful irreducibles in characteristic 13.
• Faithful irreducibles in characteristic 0.
• Dimension 13 over Z: a and b (Magma).
• Dimension 77 over Z: a and b (Magma).
• Dimension 78 over Z: a and b (Magma).
The representations of 2.A14 available are:
• Dimension 64 over GF(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 64 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 32 over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 32 over GF(49) - the automorph of the above: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 64 over GF(7) - reducible over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 64 over GF(11): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 64 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
The representations of S14 = A14:2 available are:
• All faithful primitive permutation representations.
• Permutations on 14 points - the natural representation above: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 91 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 364 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 1001 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 1716 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 2002 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 3003 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 135135 points: c and d (Magma).
• Permutations on 39916800 points - if your computer's big enough!!: c and d (Magma).
• Permutations on 39916800 points - if your computer's big enough!!: c and d (Magma).
• Faithful irreducibles in characteristic 2.
• Dimension 12 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 64 over GF(2) - a constituent of the permutation representation on 91 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 64 over GF(2) - the spin representation: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 39916800 images of the vector: v9 (Meataxe), v9 (Meataxe binary), v9 (GAP).
• Dimension 12 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
The representations of 2.S14 (plus type) available are:
The representations of 2.S14 (minus type) available are:

### 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

[Some of] The 72 conjuagcy classes of A14 are as follows:
• 1A: identity [or a3].
• 3A: a.
• 12?: ab.
• 13A: b.
• 13B: bb or b2.

[Some of] The 135 conjuagcy classes of S14 are as follows:

• 1A: identity [or c2].
• 13AB: d.
• 2D: c.
• 14D: cd. Go to main ATLAS (version 2.0) page. Go to alternating groups page. Go to old A14 page - ATLAS version 1. Anonymous ftp access is also available on sylow.mat.bham.ac.uk.

Version 2.0 created on 7th May 1999.
Last updated 19.05.99 by JNB.
Information checked to Level 0 on 19.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.