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.
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.
Black box algorithms
To find standard generators for A5:
-
Find an element x of order 2.
[The probability of success at each attempt is 1 in 4.]
-
Find an element y of order 3.
[The probability of success at each attempt is 1 in 3.]
-
Find conjugates a of x and b of y such that ab has order 5.
[The probability of success at each attempt is 2 in 5 (about 1 in 3).]
-
Now a and b are standard generators of A5.
To find standard generators for S5 = A5.2:
-
Find an element of order 6. This cubes to x in class 2B.
[The probability of success at each attempt is 1 in 6 (or 1 in 3 if you look through outer elements only).]
-
Find an element y of order 4.
[The probability of success at each attempt is 1 in 4 (or 1 in 2 if you look through outer elements only).]
-
Find conjugates c of x and d of y such that cd has order 5.
[The probability of success at each attempt is 2 in 5 (about 1 in 3).]
-
Now c and d are standard generators of S5.
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 >.
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.
Representations
The representations of A5 available are:
- All primitive permutation representations.
-
Permutations on 5 points - the natural representation above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 6 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 10 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 2.
-
Dimension 2 over GF(4) - the natural representation as L2(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 2 over GF(4) - field automorph of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 4 over GF(2) - the natural representation as O4-(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 4 over GF(2) - reducible over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 3.
-
Dimension 3 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 3 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 4 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 6 over GF(3) - reducible over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Both faithful irreducibles in characteristic 5.
-
Dimension 3 over GF(5) - the natural representation as O3(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 5 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 0.
-
Dimension 3 over Z[b5]:
a and b (Magma).
-
Dimension 3 over Z[b5]:
a and b (Magma).
-
Dimension 4 over Z:
a and b (Magma).
-
Dimension 5 over Z:
a and b (Magma).
-
Dimension 6 over Z - monomial, and reducible over Q(b5):
a and b (Magma).
The representations of 2.A5 = SL2(5) available are:
-
Permutations on 24 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 40 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- All faithful irreducibles in characteristic 3.
-
Dimension 2 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 2 over GF(9) - automorph of the above:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 4 over GF(3) - reducible over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- Both faithful irreducibles in characteristic 5.
-
Dimension 2 over GF(5) - the natural representation as SL2(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 4 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- All faithful irreducibles in characteristic 7 [not dividing the group order!!!].
-
Dimension 2 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 2 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 4 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 4 over GF(7) - reducible over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- Faithful irreducibles in characteristic 0.
-
Dimension 4 over Z[i]:
A and B (Magma).
-
Dimension 4 over Z[i6]:
A and B (Magma).
-
Dimension 8 over Z - splits as 4 + 4 over C:
A and B (Magma).
-
Dimension 6 over Z[i] - monomial:
A and B (Magma).
-
Dimension 6 over Z[w]:
A and B (Magma).
-
Dimension 12 over Z - monomial and reducible over C:
A and B (Magma).
-
Dimension 5 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 5 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 8 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 9 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 10 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of A5:2 = S5 available are:
- All faithful primitive permutation representations.
-
Permutations on 5 points - the natural representation above:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 6 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 10 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- Both faithful irreducibles in characteristic 2.
-
Dimension 4 over GF(2) - the SigmaL2(4) module:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 4 over GF(2) - the GO4-(2) module:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- Both faithful irreducibles in characteristic 3 with character in ABC.
-
Dimension 4 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 6 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- Both faithful irreducibles in characteristic 5 with character in ABC.
-
Dimension 3 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 5 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2.A5:2 = 2.S5 (plus type =
ATLAS variant) available are:
-
Permutations on 40 points - on the cosets of S3:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
.
-
Permutations on 40 points - on the cosets of C6:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 48 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- Both faithful irreducibles in characteristic 3 with character in ABC.
-
Dimension 4 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 6 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- All faithful irreducibles in characteristic 5 that correspond to
characters in the ABC.
-
Dimension 2 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 4 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 4 over GF(5) - reducible over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 8 over GF(5) - reducible over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- Faithful irreducibles in characteristic 0.
-
Dimension 4 over Z[w] - with restriction to 2A5 absolutely
irreducible:
C and D (Magma).
-
Dimension 8 over Z - with restriction to 2A5 being 2aabb, reducible over C:
C and D (Magma).
The representations of 2.A5.2 = 2.S5 (minus type =
variant not in ATLAS) available are:
-
Permutations on 48 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 80 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- All faithful irreducibles in characteristic 3 up to tensoring with
linear irreducibles.
-
Dimension 4 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 6 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 12 over GF(3) - reducible over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- All faithful irreducibles in characteristic 5 up to tensoring with
linear irreducibles.
-
Dimension 2 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 4 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 4 over GF(5) - reducible over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 8 over GF(5) - reducible over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- Faithful irreducibles in characteristic 0.
-
Dimension 4 over Z[w] - with restriction to 2A5 being 2ab - NOT YET AVAILABLE:
C and D (Magma).
-
Dimension 4 over Z[i2] - with restriction to 2A5 being 2ab:
C and D (Magma).
-
Dimension 4 over Z[i5] - with restriction to 2A5 being 2ab:
C and D (Magma).
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.
- 1A: identity [or a2].
- 2A: a.
- 3A: b.
- 5A: ab.
- 5B: (ab)2.
Representatives of the 7 conjugacy classes of S5 are given below.
- 1A: identity [or c2].
- 2A: d2.
- 3A: (cd2)2 or [c, d].
- 5AB: cd.
- 2B: c.
- 4A: d.
- 6A: cd2.
Go to main ATLAS (version 2.0) page.
Go to alternating groups page.
Go to old A5 page - ATLAS version 1.
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.