Aliases: C22⋊S5, A5⋊2D4, A5⋊C4⋊C2, (C2×S5)⋊2C2, C2.10(C2×S5), (C22×A5)⋊2C2, (C2×A5).4C22, SmallGroup(480,951)
Series: Chief►Derived ►Lower central ►Upper central
Subgroups: 1348 in 106 conjugacy classes, 9 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, D5, C10, Dic3, C12, A4, D6, C2×C6, C22⋊C4, C2×D4, C24, F5, D10, C2×C10, C4×S3, D12, C3⋊D4, C3×D4, S4, C2×A4, C22×S3, C22≀C2, C2×F5, C22×D5, A4⋊C4, S3×D4, C2×S4, C22×A4, A5, C22⋊F5, A4⋊D4, S5, C2×A5, C2×A5, A5⋊C4, C2×S5, C22×A5, C22⋊S5
Quotients: C1, C2, C22, D4, S5, C2×S5, C22⋊S5
Character table of C22⋊S5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 5 | 6A | 6B | 6C | 10A | 10B | 10C | 12 | |
| size | 1 | 1 | 2 | 15 | 15 | 20 | 30 | 20 | 20 | 60 | 60 | 24 | 20 | 40 | 40 | 24 | 24 | 24 | 40 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 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 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 0 | orthogonal lifted from D4 |
| ρ6 | 4 | 4 | -4 | 0 | 0 | 2 | 0 | 1 | -2 | 0 | 0 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | orthogonal lifted from C2×S5 |
| ρ7 | 4 | 4 | 4 | 0 | 0 | -2 | 0 | 1 | -2 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | orthogonal lifted from S5 |
| ρ8 | 4 | 4 | 4 | 0 | 0 | 2 | 0 | 1 | 2 | 0 | 0 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from S5 |
| ρ9 | 4 | 4 | -4 | 0 | 0 | -2 | 0 | 1 | 2 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×S5 |
| ρ10 | 5 | 5 | -5 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 1 | orthogonal lifted from C2×S5 |
| ρ11 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 1 | orthogonal lifted from S5 |
| ρ12 | 5 | 5 | -5 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 0 | -1 | 1 | 1 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×S5 |
| ρ13 | 5 | 5 | 5 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | -1 | orthogonal lifted from S5 |
| ρ14 | 6 | 6 | -6 | -2 | -2 | 0 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | orthogonal lifted from C2×S5 |
| ρ15 | 6 | 6 | 6 | -2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | orthogonal lifted from S5 |
| ρ16 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -√5 | √5 | -1 | 0 | orthogonal faithful |
| ρ17 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | √5 | -√5 | -1 | 0 | orthogonal faithful |
| ρ18 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | orthogonal faithful |
| ρ19 | 10 | -10 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 20 points - transitive group 20T116
Generators in S20
(1 20 5 12 4 16)(2 14)(3 18)(6 13)(7 11 9 15 8 17)(10 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
G:=sub<Sym(20)| (1,20,5,12,4,16)(2,14)(3,18)(6,13)(7,11,9,15,8,17)(10,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;
G:=Group( (1,20,5,12,4,16)(2,14)(3,18)(6,13)(7,11,9,15,8,17)(10,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );
G=PermutationGroup([[(1,20,5,12,4,16),(2,14),(3,18),(6,13),(7,11,9,15,8,17),(10,19)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)]])
G:=TransitiveGroup(20,116);
►On 20 points - transitive group 20T118
Generators in S20
(1 14 5 16 9 20)(2 8)(3 12)(4 10 6)(7 18)(11 19 15)(13 17)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
G:=sub<Sym(20)| (1,14,5,16,9,20)(2,8)(3,12)(4,10,6)(7,18)(11,19,15)(13,17), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)>;
G:=Group( (1,14,5,16,9,20)(2,8)(3,12)(4,10,6)(7,18)(11,19,15)(13,17), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20) );
G=PermutationGroup([[(1,14,5,16,9,20),(2,8),(3,12),(4,10,6),(7,18),(11,19,15),(13,17)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)]])
G:=TransitiveGroup(20,118);
►On 24 points - transitive group 24T1341
Generators in S24
(1 23 6 15 9 17)(2 18 13 20 11 22)(3 8 21 5 19 7)(4 10 16 12 24 14)
(3 4)(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,23,6,15,9,17)(2,18,13,20,11,22)(3,8,21,5,19,7)(4,10,16,12,24,14), (3,4)(5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24)>;
G:=Group( (1,23,6,15,9,17)(2,18,13,20,11,22)(3,8,21,5,19,7)(4,10,16,12,24,14), (3,4)(5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,23,6,15,9,17),(2,18,13,20,11,22),(3,8,21,5,19,7),(4,10,16,12,24,14)], [(3,4),(5,6,7,8,9),(10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,1341);
►On 24 points - transitive group 24T1342
Generators in S24
(1 16 12 23 10 20)(2 21 5 18 8 15)(3 14 19 9 17 13)(4 7 24 11 22 6)
(3 4)(5 6 7 8 9)(10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,16,12,23,10,20)(2,21,5,18,8,15)(3,14,19,9,17,13)(4,7,24,11,22,6), (3,4)(5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24)>;
G:=Group( (1,16,12,23,10,20)(2,21,5,18,8,15)(3,14,19,9,17,13)(4,7,24,11,22,6), (3,4)(5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,16,12,23,10,20),(2,21,5,18,8,15),(3,14,19,9,17,13),(4,7,24,11,22,6)], [(3,4),(5,6,7,8,9),(10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,1342);
►On 24 points - transitive group 24T1349
Generators in S24
(1 6 9 15 7 10)(2 18 14 13 12 22)(3 11 21 20 19 5)(4 23 16 8 24 17)
(1 2)(3 4)(5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,6,9,15,7,10)(2,18,14,13,12,22)(3,11,21,20,19,5)(4,23,16,8,24,17), (1,2)(3,4)(5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24)>;
G:=Group( (1,6,9,15,7,10)(2,18,14,13,12,22)(3,11,21,20,19,5)(4,23,16,8,24,17), (1,2)(3,4)(5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,6,9,15,7,10),(2,18,14,13,12,22),(3,11,21,20,19,5),(4,23,16,8,24,17)], [(1,2),(3,4),(5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,1349);
►On 24 points - transitive group 24T1350
Generators in S24
(1 19 17 21 15 23)(2 9 22 11 20 13)(3 24 7 16 5 18)(4 14 12 6 10 8)
(1 2)(3 4)(5 6 7 8 9 10 11 12 13 14)(15 16 17 18 19 20 21 22 23 24)
G:=sub<Sym(24)| (1,19,17,21,15,23)(2,9,22,11,20,13)(3,24,7,16,5,18)(4,14,12,6,10,8), (1,2)(3,4)(5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24)>;
G:=Group( (1,19,17,21,15,23)(2,9,22,11,20,13)(3,24,7,16,5,18)(4,14,12,6,10,8), (1,2)(3,4)(5,6,7,8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24) );
G=PermutationGroup([[(1,19,17,21,15,23),(2,9,22,11,20,13),(3,24,7,16,5,18),(4,14,12,6,10,8)], [(1,2),(3,4),(5,6,7,8,9,10,11,12,13,14),(15,16,17,18,19,20,21,22,23,24)]])
G:=TransitiveGroup(24,1350);
Matrix representation of C22⋊S5 ►in GL6(ℤ)
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 0 | 0 | -1 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 | 1 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | -1 | 1 | 0 |
G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,1,-1,-1,1,0,0,0,-1,0,0],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,1,1,-1,0,0,0,0,0,1,0,0,0,1,0,0] >;
C22⋊S5 in GAP, Magma, Sage, TeX
C_2^2\rtimes S_5
% in TeX
G:=Group("C2^2:S5"); // GroupNames label
G:=SmallGroup(480,951);
// by ID
G=gap.SmallGroup(480,951);
# by ID
Export