direct product, non-abelian, soluble, monomial
Aliases: C2×C4⋊S4, C23⋊2D12, C24.9D6, C4⋊2(C2×S4), (C2×C4)⋊2S4, A4⋊1(C2×D4), (C2×A4)⋊1D4, C22⋊(C2×D12), (C23×C4)⋊3S3, (C22×C4)⋊3D6, (C4×A4)⋊2C22, (C2×S4)⋊1C22, (C22×S4)⋊1C2, C2.4(C22×S4), (C2×A4).3C23, C22.24(C2×S4), C23.3(C22×S3), (C22×A4).10C22, (C2×C4×A4)⋊3C2, SmallGroup(192,1470)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4⋊S4
G = < a,b,c,d,e,f | a2=b4=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 998 in 233 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C23, C23, C12, A4, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, D12, C2×C12, S4, C2×A4, C2×A4, C22×S3, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C4×A4, C2×D12, C2×S4, C2×S4, C22×A4, C2×C4⋊D4, C4⋊S4, C2×C4×A4, C22×S4, C2×C4⋊S4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, S4, C22×S3, C2×D12, C2×S4, C4⋊S4, C22×S4, C2×C4⋊S4
Character table of C2×C4⋊S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 12 | 12 | 12 | 12 | 8 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | -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 | -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 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ11 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ12 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ19 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ20 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 3 | 3 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ21 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | -3 | -3 | 1 | 1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ22 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 0 | -3 | -3 | 1 | 1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ23 | 3 | 3 | -3 | -3 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 0 | -3 | 3 | -1 | 1 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ24 | 3 | 3 | -3 | -3 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 | -3 | 3 | -1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ25 | 3 | 3 | -3 | -3 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 0 | 3 | -3 | 1 | -1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ26 | 3 | 3 | -3 | -3 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 0 | 3 | -3 | 1 | -1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ27 | 6 | -6 | 6 | -6 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4⋊S4 |
| ρ28 | 6 | -6 | -6 | 6 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C4⋊S4 |
►On 24 points - transitive group 24T394
Generators in S24
(1 8)(2 5)(3 6)(4 7)(9 16)(10 13)(11 14)(12 15)(17 24)(18 21)(19 22)(20 23)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 6)(2 7)(3 8)(4 5)(17 22)(18 23)(19 24)(20 21)
(9 14)(10 15)(11 16)(12 13)(17 22)(18 23)(19 24)(20 21)
(1 23 9)(2 24 10)(3 21 11)(4 22 12)(5 17 13)(6 18 14)(7 19 15)(8 20 16)
(1 6)(2 5)(3 8)(4 7)(9 18)(10 17)(11 20)(12 19)(13 24)(14 23)(15 22)(16 21)
G:=sub<Sym(24)| (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15)(17,24)(18,21)(19,22)(20,23), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,7)(3,8)(4,5)(17,22)(18,23)(19,24)(20,21), (9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,23,9)(2,24,10)(3,21,11)(4,22,12)(5,17,13)(6,18,14)(7,19,15)(8,20,16), (1,6)(2,5)(3,8)(4,7)(9,18)(10,17)(11,20)(12,19)(13,24)(14,23)(15,22)(16,21)>;
G:=Group( (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15)(17,24)(18,21)(19,22)(20,23), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,6)(2,7)(3,8)(4,5)(17,22)(18,23)(19,24)(20,21), (9,14)(10,15)(11,16)(12,13)(17,22)(18,23)(19,24)(20,21), (1,23,9)(2,24,10)(3,21,11)(4,22,12)(5,17,13)(6,18,14)(7,19,15)(8,20,16), (1,6)(2,5)(3,8)(4,7)(9,18)(10,17)(11,20)(12,19)(13,24)(14,23)(15,22)(16,21) );
G=PermutationGroup([[(1,8),(2,5),(3,6),(4,7),(9,16),(10,13),(11,14),(12,15),(17,24),(18,21),(19,22),(20,23)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,6),(2,7),(3,8),(4,5),(17,22),(18,23),(19,24),(20,21)], [(9,14),(10,15),(11,16),(12,13),(17,22),(18,23),(19,24),(20,21)], [(1,23,9),(2,24,10),(3,21,11),(4,22,12),(5,17,13),(6,18,14),(7,19,15),(8,20,16)], [(1,6),(2,5),(3,8),(4,7),(9,18),(10,17),(11,20),(12,19),(13,24),(14,23),(15,22),(16,21)]])
G:=TransitiveGroup(24,394);
►On 24 points - transitive group 24T419
Generators in S24
(1 23)(2 24)(3 21)(4 22)(5 15)(6 16)(7 13)(8 14)(9 18)(10 19)(11 20)(12 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)
(1 3)(2 4)(5 7)(6 8)(13 15)(14 16)(21 23)(22 24)
(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)
(1 13 11)(2 14 12)(3 15 9)(4 16 10)(5 18 21)(6 19 22)(7 20 23)(8 17 24)
(1 22)(2 21)(3 24)(4 23)(5 12)(6 11)(7 10)(8 9)(13 19)(14 18)(15 17)(16 20)
G:=sub<Sym(24)| (1,23)(2,24)(3,21)(4,22)(5,15)(6,16)(7,13)(8,14)(9,18)(10,19)(11,20)(12,17), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,13,11)(2,14,12)(3,15,9)(4,16,10)(5,18,21)(6,19,22)(7,20,23)(8,17,24), (1,22)(2,21)(3,24)(4,23)(5,12)(6,11)(7,10)(8,9)(13,19)(14,18)(15,17)(16,20)>;
G:=Group( (1,23)(2,24)(3,21)(4,22)(5,15)(6,16)(7,13)(8,14)(9,18)(10,19)(11,20)(12,17), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3)(2,4)(5,7)(6,8)(13,15)(14,16)(21,23)(22,24), (5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20), (1,13,11)(2,14,12)(3,15,9)(4,16,10)(5,18,21)(6,19,22)(7,20,23)(8,17,24), (1,22)(2,21)(3,24)(4,23)(5,12)(6,11)(7,10)(8,9)(13,19)(14,18)(15,17)(16,20) );
G=PermutationGroup([[(1,23),(2,24),(3,21),(4,22),(5,15),(6,16),(7,13),(8,14),(9,18),(10,19),(11,20),(12,17)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3),(2,4),(5,7),(6,8),(13,15),(14,16),(21,23),(22,24)], [(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20)], [(1,13,11),(2,14,12),(3,15,9),(4,16,10),(5,18,21),(6,19,22),(7,20,23),(8,17,24)], [(1,22),(2,21),(3,24),(4,23),(5,12),(6,11),(7,10),(8,9),(13,19),(14,18),(15,17),(16,20)]])
G:=TransitiveGroup(24,419);
Matrix representation of C2×C4⋊S4 ►in GL7(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 6 | 0 | 0 | 0 |
| 0 | 0 | 8 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(7,GF(13))| [12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,9,8,0,0,0,0,0,6,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,1,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,11,1,12],[12,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,11,1,12,0,0,0,0,0,0,12] >;
C2×C4⋊S4 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes S_4
% in TeX
G:=Group("C2xC4:S4"); // GroupNames label
G:=SmallGroup(192,1470);
// by ID
G=gap.SmallGroup(192,1470);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,2,254,58,1124,4037,285,2358,475]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^4=c^2=d^2=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=f*c*f=c*d=d*c,e*d*e^-1=c,d*f=f*d,f*e*f=e^-1>;
// generators/relations
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