ATLAS: Alternating group A_{14}
Order = 43589145600 = 2^{10}.3^{5}.5^{2}.7^{2}.11.13.
Mult = 2.
Out = 2.
Standard generators
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.
Automorphisms
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.
Representations
The representations of A_{14} 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.A_{14} 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 S_{14} = A_{14}: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.S_{14} (plus type) available are:
 C and
D as
64 × 64 matrices over GF(7).
The representations of 2.S_{14} (minus type) available are:
Maximal subgroups
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:
Conjugacy classes
The 72 conjuagcy classes of A_{14} are as follows:
Class 
Cycle type 
p power 
p' part 
Order of centraliser 
Centraliser 
Representative 
1A 
1^{14} 
 
 
43589145600 
A_{14} 
identity 
2A 
1^{10}.2^{2} 
A 
A 
14515200 
(A_{10} × 2^{2}):2 
[a, b] 
2B 
1^{6}.2^{4} 
A 
A 
138240 


2C 
1^{2}.2^{6} 
A 
A 
46080 


3A 
1^{11}.3 
A 
A 
59875200 
A_{11} × 3 
a 
3B 
1^{8}.3^{2} 
A 
A 
362880 
(A_{8} × 3^{2}):2 

3C 
1^{5}.3^{3} 
A 
A 
9720 


3D 
1^{2}.3^{4} 
A 
A 
1944 


4A 
1^{8}.2.4 
A 
A 
161280 


4B 
1^{6}.4^{2} 
B 
A 
11520 


4C 
2^{5}.4 
A 
A 
7680 


4D 
1^{4}.2^{3}.4 
A 
A 
2304 


4E 
2.4^{3} 
C 
A 
384 


4F 
1^{2}.2^{2}.4^{2} 
B 
A 
256 


5A 
1^{9}.5 
A 
A 
907200 
A_{9} × 5 

5B 
1^{4}.5^{2} 
A 
A 
600 
(A_{4} × 5^{2}):2 








21A 
1^{4}.3.7 
AA 
AA 
252 
A_{4} × 21 

21B 
1.3^{2}.7 
AB 
AB 
63 
3^{2} × 7 

24A 
1.2.3.8 


24 
24 

24B 
6.8 


24 
24 
ab^{2} or ab^{5} 
28A 
1.2.4.7 
AA 
AA 
28 
28 

30A 
1^{2}.2^{2}.3.5 


120 
15 × D_{8} 

30B 
1.2.5.6 


30 
30 

33A 
3.11 
AA 
AA 
33 
33 

33B* 
3.11 
AA 
AA 
33 
33 
ab^{4} or ab^{6} 
35A 
1^{2}.5.7 
AA 
AA 
35 
35 

42A 
2^{2}.3.7 


84 
21 × 2^{2} 

45A 
5.9 


45 
45 
ab^{5} or ab^{3} 
45B* 
5.9 


45 
45 

60A 
2.3.4.5 


60 
60 

The 135 conjuagcy classes of S_{14} are as follows:
Class 
Cycle type 
Order of centraliser 
Centraliser 
Representative 
2A  1^{10}.2^{2} 
2B  1^{6}.2^{4} 
2C  1^{2}.2^{6} 
Return to main ATLAS page.
Last updated 12th May 1998,
R.A.Wilson, R.A.Parker and J.N.Bray