ATLAS: Alternating group A_{14}
Order = 43589145600 = 2^{10}.3^{5}.5^{2}.7^{2}.11.13.
Mult = 2.
Out = 2.
The following information is available for A_{14}:
Standard generators of A_{14} 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.A_{14} are preimages
A and B where A has order 3 and B has order 13.
Standard generators of S_{14} = A_{14}: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.S_{14} are
preimages C and D where D has order 13.
In the natural representations given here, we have a = cd^{1}cd = [c, d] and b = d.
An outer automorphism of A_{14} 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.
To find standard generators for A_{14}:

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 A_{14}.
To find standard generators for S_{14} = A_{14}.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 S_{14}.
Presentations for A_{14} and S_{14} (respectively) on their standard generators are given below.
< a, b  a^{3} = b^{13} = (ab)^{12} = [a, b]^{2} = (aa^{bab})^{2} = [a, babab]^{2} = (aa^{bababab})^{2} = [a, babababab]^{2} = 1 >.
< c, d  c^{2} = d^{13} = (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 are available for the following decorations of A_{14}.
The representations of A_{14} 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.

Dimension 13 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 64 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Faithful irreducibles in characteristic 5.

Dimension 13 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Faithful irreducibles in characteristic 7.

Dimension 12 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 66 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Faithful irreducibles in characteristic 11.

Dimension 13 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Faithful irreducibles in characteristic 13.

Dimension 13 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 76 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 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.A_{14} 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 S_{14} = A_{14}: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.S_{14} (plus type) available are:

Dimension 64 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 2.S_{14} (minus type) available are:

Dimension 64 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of A_{14} are as follows:
 A_{13}.
Order: 3113510400.
Index: 14.
 S_{12} = A_{12}:2.
Order: 479001600.
Index: 91.
 (A_{11} × 3):2.
Order: 119750400.
Index: 364.
 (A_{10} × A_{4}):2.
Order: 43545600.
Index: 1001.
 (A_{7} × A_{7}):4.
Order: 25401600.
Index: 1716.
 (A_{9} × A_{5}):2.
Order: 21772800.
Index: 2002.
 (A_{8} × A_{6}):2_{1}.
Order: 14515200.
Index: 3003.
 2^{6}:S_{7}.
Order: 322560.
Index: 135135.
 L_{2}(13).
Order: 1092.
Index: 39916800.
The maximal subgroups of S_{14} are as follows:
[Some of] The 72 conjuagcy classes of A_{14} are as follows:
 1A: identity [or a^{3}].
 3A: a.^{ }
 12?: ab.^{ }
 13A: b.^{ }
 13B: bb or b^{2}.
[Some of] The 135 conjuagcy classes of S_{14} are as follows:
 1A: identity [or c^{2}].
 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.