direct product, non-abelian, soluble, monomial, rational
Aliases: C2×C3⋊S4, C6⋊S4, A4⋊2D6, (C2×A4)⋊S3, C3⋊2(C2×S4), (C2×C6)⋊3D6, (C6×A4)⋊2C2, C23⋊(C3⋊S3), (C22×C6)⋊2S3, (C3×A4)⋊3C22, C22⋊(C2×C3⋊S3), SmallGroup(144,189)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×A4 — C2×C3⋊S4 |
Generators and relations for C2×C3⋊S4
G = < a,b,c,d,e,f | a2=b3=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: 450 in 86 conjugacy classes, 19 normal (11 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, A4, D6, C2×C6, C2×C6, C2×D4, C3⋊S3, C3×C6, C2×Dic3, C3⋊D4, S4, C2×A4, C22×S3, C22×C6, C3×A4, C2×C3⋊S3, C2×C3⋊D4, C2×S4, C3⋊S4, C6×A4, C2×C3⋊S4
Quotients: C1, C2, C22, S3, D6, C3⋊S3, S4, C2×C3⋊S3, C2×S4, C3⋊S4, C2×C3⋊S4
Character table of C2×C3⋊S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 3 | 3 | 18 | 18 | 2 | 8 | 8 | 8 | 18 | 18 | 2 | 6 | 6 | 8 | 8 | 8 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | 1 | -1 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ6 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | -2 | 2 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | 1 | -1 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 0 | 1 | -1 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ13 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | 0 | 1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ14 | 3 | 3 | -1 | -1 | 1 | 1 | 3 | 0 | 0 | 0 | -1 | -1 | 3 | -1 | -1 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | -3 | -1 | 1 | -1 | 1 | 3 | 0 | 0 | 0 | 1 | -1 | -3 | -1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | -3 | -1 | 1 | 1 | -1 | 3 | 0 | 0 | 0 | -1 | 1 | -3 | -1 | 1 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ17 | 6 | 6 | -2 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | -3 | 1 | 1 | 0 | 0 | 0 | orthogonal lifted from C3⋊S4 |
| ρ18 | 6 | -6 | -2 | 2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | -1 | 0 | 0 | 0 | orthogonal faithful |
►On 18 points - transitive group 18T66
Generators in S18
(1 7)(2 8)(3 9)(4 16)(5 17)(6 18)(10 13)(11 14)(12 15)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 7)(2 8)(3 9)(10 13)(11 14)(12 15)
(4 16)(5 17)(6 18)(10 13)(11 14)(12 15)
(1 10 16)(2 11 17)(3 12 18)(4 7 13)(5 8 14)(6 9 15)
(2 3)(4 13)(5 15)(6 14)(8 9)(10 16)(11 18)(12 17)
G:=sub<Sym(18)| (1,7)(2,8)(3,9)(4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15), (4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,10,16)(2,11,17)(3,12,18)(4,7,13)(5,8,14)(6,9,15), (2,3)(4,13)(5,15)(6,14)(8,9)(10,16)(11,18)(12,17)>;
G:=Group( (1,7)(2,8)(3,9)(4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,7)(2,8)(3,9)(10,13)(11,14)(12,15), (4,16)(5,17)(6,18)(10,13)(11,14)(12,15), (1,10,16)(2,11,17)(3,12,18)(4,7,13)(5,8,14)(6,9,15), (2,3)(4,13)(5,15)(6,14)(8,9)(10,16)(11,18)(12,17) );
G=PermutationGroup([[(1,7),(2,8),(3,9),(4,16),(5,17),(6,18),(10,13),(11,14),(12,15)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,7),(2,8),(3,9),(10,13),(11,14),(12,15)], [(4,16),(5,17),(6,18),(10,13),(11,14),(12,15)], [(1,10,16),(2,11,17),(3,12,18),(4,7,13),(5,8,14),(6,9,15)], [(2,3),(4,13),(5,15),(6,14),(8,9),(10,16),(11,18),(12,17)]])
G:=TransitiveGroup(18,66);
►On 24 points - transitive group 24T251
Generators in S24
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 20)
(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 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)
(1 18)(2 16)(3 17)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(19 24)(20 22)(21 23)
(1 2 3)(4 5 6)(7 22 21)(8 23 19)(9 24 20)(10 13 17)(11 14 18)(12 15 16)
(2 3)(5 6)(7 9)(10 15)(11 14)(12 13)(16 17)(19 23)(20 22)(21 24)
G:=sub<Sym(24)| (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (1,18)(2,16)(3,17)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(19,24)(20,22)(21,23), (1,2,3)(4,5,6)(7,22,21)(8,23,19)(9,24,20)(10,13,17)(11,14,18)(12,15,16), (2,3)(5,6)(7,9)(10,15)(11,14)(12,13)(16,17)(19,23)(20,22)(21,24)>;
G:=Group( (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (1,18)(2,16)(3,17)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(19,24)(20,22)(21,23), (1,2,3)(4,5,6)(7,22,21)(8,23,19)(9,24,20)(10,13,17)(11,14,18)(12,15,16), (2,3)(5,6)(7,9)(10,15)(11,14)(12,13)(16,17)(19,23)(20,22)(21,24) );
G=PermutationGroup([[(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,15),(2,13),(3,14),(4,20),(5,21),(6,19),(7,24),(8,22),(9,23),(10,18),(11,16),(12,17)], [(1,18),(2,16),(3,17),(4,8),(5,9),(6,7),(10,15),(11,13),(12,14),(19,24),(20,22),(21,23)], [(1,2,3),(4,5,6),(7,22,21),(8,23,19),(9,24,20),(10,13,17),(11,14,18),(12,15,16)], [(2,3),(5,6),(7,9),(10,15),(11,14),(12,13),(16,17),(19,23),(20,22),(21,24)]])
G:=TransitiveGroup(24,251);
C2×C3⋊S4 is a maximal subgroup of
Dic3⋊2S4 Dic3⋊S4 A4⋊D12 C12⋊S4 (C2×C6)⋊4S4 C2×S3×S4
C2×C3⋊S4 is a maximal quotient of A4⋊Dic6 C12⋊S4 SL2(𝔽3).D6 C12.6S4 C12.14S4 C12.7S4 (C2×C6)⋊4S4
Matrix representation of C2×C3⋊S4 ►in GL5(ℤ)
| -1 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| -1 | 1 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | -1 | 0 | 0 | 0 |
| 1 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 0 | 0 |
| -1 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 |
G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,-1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1],[0,1,0,0,0,-1,-1,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,-1,0],[-1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,-1] >;
C2×C3⋊S4 in GAP, Magma, Sage, TeX
C_2\times C_3\rtimes S_4
% in TeX
G:=Group("C2xC3:S4"); // GroupNames label
G:=SmallGroup(144,189);
// by ID
G=gap.SmallGroup(144,189);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-2,2,146,579,2164,556,1301,989]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=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
Export