ATLAS: Alternating group A5

Order = 60 = 22.3.5.
Mult = 2.
Out = 2.

Standard generators

Standard generators of A5 are a and b where a has order 2, b has order 3 and ab has order 5.
In the natural representation we may take a = (1, 2)(3, 4) and b = (1, 3, 5).
Standard generators of the double cover 2.A5 (or SL2(5)) are preimages A and B where B has order 3 and AB has order 5.

Standard generators of the automorphism group S5 = A5:2 are c and d where c is in class 2B, d has order 4 and cd has order 5.
In the natural representation we may take c = (1, 2) and d = (2, 3, 4, 5).
Standard generators of a double cover 2.S5 (containing SL2(5) to index 2) are preimages C and D where CD has order 5.


Automorphisms

An outer automorphism of A5 is given by (a, b) maps to (a, abbababb), which corresponds to the transposition (3, 4) of S5 if you take the same generators of A5 as above.

If u is the above automorphism, then we have c = (ab)2u(ab)-2 and d = (ab)-2u(ab)-2 = abc.
The pair (c', d') is conjugate in S5 to (c, d) where c' = u and d' = uab.

Conversely, we have a = [c, dcd] and b = (dcd)-2.

Please note that (a, b) -> (a, ababbab) is also an outer automorphism of A5, but in S5 = Aut(A5) this element has order 4 and squares to a.


Presentations

Presentations for A5 and S5 (respectively) on their standard generators are given below.

< a, b | a2 = b3 = (ab)5 = 1 >.

< c, d | c2 = d4 = (cd)5 = [c, d]3 = 1 >.


Representations

The representations of A5 available are as follows (in the table below).
Degree Type/Ring Generators Name Info Comments (if any)
All primitive permutation representations
5 permutation a b A5p5 - The natural representation of A5
6 permutation a b A5p6 - -
10 permutation a b A5p10 - -
All faithful irreducibles in characteristic 2 and over GF(2)
2 GF(4) a b A5f4 - The natural representation as L2(4)
2 GF(4) a b A5f4b - -
4 GF(2) a b A5f2 - The natural representation as O4-(2)
4 GF(2) a b A5f2r2ab - Reducible over GF(4)
All faithful irreducibles in characteristic 3 and over GF(3)
3 GF(9) a b A5f9 - -
3 GF(9) a b A5f9b - -
4 GF(3) a b A5f3 - -
6 GF(3) a b A5f3r6 - Reducible over GF(9)
All faithful irreducibles in characteristic 5
3 GF(5) a b A5f5 - The natural representation as O3(5)
5 GF(5) a b A5f5r5 - -
All faithful irreducibles in characteristic 0 and over Z
3 Z[b5] a and b A5Ar3a - -
3 Z[b5] a and b A5Ar3b - -
4 Z a and b A5Zr4 - -
5 Z a and b A5Zr5 - -
6 Z a and b A5Zr6 - Monomial, and reducible over Q(b5)

The representations of 2.A5 = SL2(5) available are

The representations of A5:2 = S5 available are: The representations of 2.A5:2 = 2.S5 (plus type) available are:

Maximal subgroups

The maximal subgroups of A5 are as follows.
Class Order Index Specification Generators
A4 12 5 N(2A2) ababba, b
D10 10 6 N(5AB) a, bbab
S3 6 10 N(3A) a, ababbabab
The maximal subgroups of S5 are as follows.

Conjugacy classes

The 5 conjugacy classes of A5 are as follows:
Class Cycle type p power p' part Order of centraliser Centraliser Representative
1A 15 - - 60 A5 identity
2A 1.22 A A 4 22 a
3A 12.3 A A 3 3 b
5A 5 A A 5 5 ab
5B* 5 A A 5 5 (ab)2

The 7 conjugacy classes of S5 are as follows:
Class Cycle type p power p' part Order of centraliser Centraliser Representative
1A 15 - - 120 S5 identity
2A 1.22 A A 8 D8 d2
3A 12.3 A A 6 6 (cd2)2 or [c, d]
5AB 5 A A 5 5 cd
2B 13.2 A A 12 D12 c
4A 1.4 A A 4 4 d
6A 2.3 AB AB 6 6 cd2

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Last update14th June 1999,
R.A.Wilson, R.A.Parker and J.N.Bray