ATLAS: Alternating group A5, Linear groups L2(4) and L2(5)

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

The following information is available for A5 = L2(4) = L2(5):

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 either of the double covers 2.S5 (containing SL2(5) to index 2) are preimages C and D where CD has order 5.


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.

Black box algorithms

To find standard generators for A5: To find standard generators for S5 = A5.2:


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 >.

These presentations, and those of the covering groups, are available in Magma format as follows:
A5 on a and b, 2A5 on A and B, S5 on c and d, 2S5 (+) on C and D and 2S5 (-) on C and D.


The representations of A5 available are: 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 = ATLAS variant) available are: The representations of 2.A5.2 = 2.S5 (minus type = variant not in ATLAS) available are:

Maximal subgroups

The maximal subgroups of A5 are as follows. The maximal subgroups of S5 are as follows.

Conjugacy classes

Representatives of the 5 conjugacy classes of A5 are given below.
Representatives of the 7 conjugacy classes of S5 are given below.
Main ATLAS page Go to main ATLAS (version 2.0) page.
Alternating groups page Go to alternating groups page.
Old A5 page Go to old A5 page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 23rd April 1999.
Last updated 13.12.01 by JNB.
Information checked to Level 1 on 08.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.