Aliases: A5⋊C4, C2.1S5, (C2×A5).C2, SmallGroup(240,91)
Series: Chief►Derived ►Lower central ►Upper central
| A5 — A5⋊C4 |
| A5 — A5⋊C4 |
15C2
15C2
10C3
6C5
5C22
10C4
15C22
15C22
30C4
10C6
10S3
10S3
6D5
6C10
6D5
5C23
15C2×C4
15C2×C4
5A4
10Dic3
10D6
10C12
6F5
6D10
6F5
15C22⋊C4
10C4×S3
Character table of A5⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 5 | 6 | 10 | 12A | 12B | |
| size | 1 | 1 | 15 | 15 | 20 | 10 | 10 | 30 | 30 | 24 | 20 | 24 | 20 | 20 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | -1 | -1 | 1 | 1 | i | -i | -i | i | 1 | -1 | -1 | -i | i | linear of order 4 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | -i | i | i | -i | 1 | -1 | -1 | i | -i | linear of order 4 |
| ρ5 | 4 | 4 | 0 | 0 | 1 | 2 | 2 | 0 | 0 | -1 | 1 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ6 | 4 | 4 | 0 | 0 | 1 | -2 | -2 | 0 | 0 | -1 | 1 | -1 | 1 | 1 | orthogonal lifted from S5 |
| ρ7 | 4 | -4 | 0 | 0 | 1 | -2i | 2i | 0 | 0 | -1 | -1 | 1 | -i | i | complex faithful |
| ρ8 | 4 | -4 | 0 | 0 | 1 | 2i | -2i | 0 | 0 | -1 | -1 | 1 | i | -i | complex faithful |
| ρ9 | 5 | 5 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 0 | -1 | 0 | -1 | -1 | orthogonal lifted from S5 |
| ρ10 | 5 | 5 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 0 | -1 | 0 | 1 | 1 | orthogonal lifted from S5 |
| ρ11 | 5 | -5 | -1 | 1 | -1 | i | -i | i | -i | 0 | 1 | 0 | -i | i | complex faithful |
| ρ12 | 5 | -5 | -1 | 1 | -1 | -i | i | -i | i | 0 | 1 | 0 | i | -i | complex faithful |
| ρ13 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | -1 | 0 | 0 | orthogonal faithful |
| ρ14 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | orthogonal lifted from S5 |
►On 12 points - transitive group 12T124
Generators in S12
(3 4 5 6 7)(8 9 10 11 12)
(1 10 2 5)(3 6 8 11)(4 12 9 7)
G:=sub<Sym(12)| (3,4,5,6,7)(8,9,10,11,12), (1,10,2,5)(3,6,8,11)(4,12,9,7)>;
G:=Group( (3,4,5,6,7)(8,9,10,11,12), (1,10,2,5)(3,6,8,11)(4,12,9,7) );
G=PermutationGroup([[(3,4,5,6,7),(8,9,10,11,12)], [(1,10,2,5),(3,6,8,11),(4,12,9,7)]])
G:=TransitiveGroup(12,124);
►On 20 points - transitive group 20T66
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 19 11 10)(2 18 12 9)(3 16 13 7)(4 17 14 8)(5 20 15 6)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,19,11,10)(2,18,12,9)(3,16,13,7)(4,17,14,8)(5,20,15,6)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,19,11,10)(2,18,12,9)(3,16,13,7)(4,17,14,8)(5,20,15,6) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,19,11,10),(2,18,12,9),(3,16,13,7),(4,17,14,8),(5,20,15,6)]])
G:=TransitiveGroup(20,66);
►On 24 points - transitive group 24T571
Generators in S24
(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19)(20 21 22 23 24)
(1 20 3 6)(2 13 4 17)(5 10 24 19)(7 11 21 15)(8 12 22 16)(9 14 23 18)
G:=sub<Sym(24)| (5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19)(20,21,22,23,24), (1,20,3,6)(2,13,4,17)(5,10,24,19)(7,11,21,15)(8,12,22,16)(9,14,23,18)>;
G:=Group( (5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19)(20,21,22,23,24), (1,20,3,6)(2,13,4,17)(5,10,24,19)(7,11,21,15)(8,12,22,16)(9,14,23,18) );
G=PermutationGroup([[(5,6,7,8,9),(10,11,12,13,14),(15,16,17,18,19),(20,21,22,23,24)], [(1,20,3,6),(2,13,4,17),(5,10,24,19),(7,11,21,15),(8,12,22,16),(9,14,23,18)]])
G:=TransitiveGroup(24,571);
►On 24 points - transitive group 24T578
Generators in S24
(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19)(20 21 22 23 24)
(1 13 3 19)(2 7 4 20)(5 15 23 14)(6 10 24 16)(8 11 21 17)(9 18 22 12)
G:=sub<Sym(24)| (5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19)(20,21,22,23,24), (1,13,3,19)(2,7,4,20)(5,15,23,14)(6,10,24,16)(8,11,21,17)(9,18,22,12)>;
G:=Group( (5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19)(20,21,22,23,24), (1,13,3,19)(2,7,4,20)(5,15,23,14)(6,10,24,16)(8,11,21,17)(9,18,22,12) );
G=PermutationGroup([[(5,6,7,8,9),(10,11,12,13,14),(15,16,17,18,19),(20,21,22,23,24)], [(1,13,3,19),(2,7,4,20),(5,15,23,14),(6,10,24,16),(8,11,21,17),(9,18,22,12)]])
G:=TransitiveGroup(24,578);
A5⋊C4 is a maximal subgroup of
C4×S5 A5⋊Q8 C22⋊S5
A5⋊C4 is a maximal quotient of A5⋊C8 GL2(𝔽5) C22.2S5
| action | f(x) | Disc(f) |
|---|---|---|
| 12T124 | x12-6x11+14x10-15x9+5x8+4x7-4x6+x5-4x2+4x-1 | 515·134 |
Matrix representation of A5⋊C4 ►in GL3(𝔽5) generated by
| 0 | 3 | 1 |
| 1 | 1 | 3 |
| 0 | 2 | 2 |
| 3 | 1 | 2 |
| 1 | 3 | 2 |
| 2 | 2 | 1 |
G:=sub<GL(3,GF(5))| [0,1,0,3,1,2,1,3,2],[3,1,2,1,3,2,2,2,1] >;
A5⋊C4 in GAP, Magma, Sage, TeX
A_5\rtimes C_4
% in TeX
G:=Group("A5:C4"); // GroupNames label
G:=SmallGroup(240,91);
// by ID
G=gap.SmallGroup(240,91);
# by ID
Export
Subgroup lattice of A5⋊C4 in TeX
Character table of A5⋊C4 in TeX