Standard generators of the automorphism group S_{5} = A_{5}: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.S_{5} (containing SL_{2}(5) to index 2) are preimages
C and D where
CD has order 5.
If u is the above automorphism, then we have c = (ab)^{2}u(ab)^{-2} and d = (ab)^{-2}u(ab)^{-2} = abc.
The pair (c', d') is conjugate in S_{5} 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 A_{5}, but in S_{5} = Aut(A_{5}) this element has order 4 and squares to a.
< a, b | a^{2} = b^{3} = (ab)^{5} = 1 >.
< c, d | c^{2} = d^{4} = (cd)^{5} = [c, d]^{3} = 1 >.
Degree | Type/Ring | Generators | Name | Info | Comments (if any) | ||
---|---|---|---|---|---|---|---|
All primitive permutation representations | |||||||
5 | permutation | a | b | A5p5 | - | The natural representation of A_{5} | |
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 L_{2}(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 O_{3}(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
Class | Order | Index | Specification | Generators |
---|---|---|---|---|
A_{4} | 12 | 5 | N(2A^{2}) | ababba, b |
D_{10} | 10 | 6 | N(5AB) | a, bbab |
S_{3} | 6 | 10 | N(3A) | a, ababbabab |
Class | Cycle type | p power | p' part | Order of centraliser | Centraliser | Representative |
---|---|---|---|---|---|---|
1A | 1^{5} | - | - | 60 | A_{5} | identity |
2A | 1.2^{2} | A | A | 4 | 2^{2} | a |
3A | 1^{2}.3 | A | A | 3 | 3 | b |
5A | 5 | A | A | 5 | 5 | ab |
5B* | 5 | A | A | 5 | 5 | (ab)^{2} |
Class | Cycle type | p power | p' part | Order of centraliser | Centraliser | Representative |
---|---|---|---|---|---|---|
1A | 1^{5} | - | - | 120 | S_{5} | identity |
2A | 1.2^{2} | A | A | 8 | D_{8} | d^{2} |
3A | 1^{2}.3 | A | A | 6 | 6 | (cd^{2})^{2} or [c, d] |
5AB | 5 | A | A | 5 | 5 | cd |
2B | 1^{3}.2 | A | A | 12 | D_{12} | c |
4A | 1.4 | A | A | 4 | 4 | d |
6A | 2.3 | AB | AB | 6 | 6 | cd^{2} |
Last update14th June 1999,