direct product, metabelian, soluble, monomial, A-group
Aliases: C2xS3xA4, C6:(C2xA4), C3:(C22xA4), (C22xC6):C6, (C6xA4):3C2, (S3xC23):C3, (C22xS3):C6, C22:2(S3xC6), C23:2(C3xS3), (C3xA4):4C22, (C2xC6):(C2xC6), SmallGroup(144,190)
Series: Derived ►Chief ►Lower central ►Upper central
| C2xC6 — C2xS3xA4 |
Generators and relations for C2xS3xA4
G = < a,b,c,d,e,f | a2=b3=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >
Subgroups: 336 in 82 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C3, C3, C22, C22, S3, S3, C6, C6, C23, C23, C32, A4, A4, D6, D6, C2xC6, C2xC6, C24, C3xS3, C3xC6, C2xA4, C2xA4, C22xS3, C22xS3, C22xC6, C3xA4, S3xC6, C22xA4, S3xC23, S3xA4, C6xA4, C2xS3xA4
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2xC6, C3xS3, C2xA4, S3xC6, C22xA4, S3xA4, C2xS3xA4
Character table of C2xS3xA4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | |
| size | 1 | 1 | 3 | 3 | 3 | 3 | 9 | 9 | 2 | 4 | 4 | 8 | 8 | 2 | 4 | 4 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | -1 | ζ6 | ζ65 | 1 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | ζ3 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ6 | linear of order 6 |
| ρ8 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | -1 | ζ6 | ζ65 | 1 | -1 | ζ65 | ζ6 | ζ32 | ζ3 | ζ6 | ζ65 | linear of order 6 |
| ρ9 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | -1 | ζ65 | ζ6 | 1 | -1 | ζ6 | ζ65 | ζ3 | ζ32 | ζ65 | ζ6 | linear of order 6 |
| ρ10 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ65 | linear of order 6 |
| ρ11 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | -1 | ζ65 | ζ6 | 1 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | ζ32 | linear of order 6 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ13 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 1 | -2 | -2 | -1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ15 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 1 | 1-√-3 | 1+√-3 | -1 | 1 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | complex lifted from S3xC6 |
| ρ16 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 1 | 1+√-3 | 1-√-3 | -1 | 1 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | complex lifted from S3xC6 |
| ρ17 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1 | -1-√-3 | -1+√-3 | -1 | -1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ18 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1 | -1+√-3 | -1-√-3 | -1 | -1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | complex lifted from C3xS3 |
| ρ19 | 3 | -3 | 3 | 1 | -1 | -3 | -1 | 1 | 3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ20 | 3 | 3 | -3 | -1 | -1 | -3 | 1 | 1 | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ21 | 3 | 3 | 3 | -1 | -1 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ22 | 3 | -3 | -3 | 1 | -1 | 3 | 1 | -1 | 3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xA4 |
| ρ23 | 6 | -6 | 0 | 2 | -2 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ24 | 6 | 6 | 0 | -2 | -2 | 0 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3xA4 |
►On 18 points - transitive group 18T60
Generators in S18
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 4)(2 6)(3 5)(7 10)(8 12)(9 11)(13 16)(14 18)(15 17)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)
(1 4)(2 5)(3 6)(13 16)(14 17)(15 18)
(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)
G:=sub<Sym(18)| (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,4)(2,6)(3,5)(7,10)(8,12)(9,11)(13,16)(14,18)(15,17), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)>;
G:=Group( (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,4)(2,6)(3,5)(7,10)(8,12)(9,11)(13,16)(14,18)(15,17), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (1,4)(2,5)(3,6)(13,16)(14,17)(15,18), (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12) );
G=PermutationGroup([[(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,4),(2,6),(3,5),(7,10),(8,12),(9,11),(13,16),(14,18),(15,17)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12)], [(1,4),(2,5),(3,6),(13,16),(14,17),(15,18)], [(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12)]])
G:=TransitiveGroup(18,60);
►On 24 points - transitive group 24T250
Generators in S24
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(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 13)(2 15)(3 14)(4 16)(5 18)(6 17)(7 19)(8 21)(9 20)(10 22)(11 24)(12 23)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(4 7 10)(5 8 11)(6 9 12)(16 19 22)(17 20 23)(18 21 24)
G:=sub<Sym(24)| (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,13)(2,15)(3,14)(4,16)(5,18)(6,17)(7,19)(8,21)(9,20)(10,22)(11,24)(12,23), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (4,7,10)(5,8,11)(6,9,12)(16,19,22)(17,20,23)(18,21,24)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24), (1,13)(2,15)(3,14)(4,16)(5,18)(6,17)(7,19)(8,21)(9,20)(10,22)(11,24)(12,23), (1,10)(2,11)(3,12)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (4,7,10)(5,8,11)(6,9,12)(16,19,22)(17,20,23)(18,21,24) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24)], [(1,13),(2,15),(3,14),(4,16),(5,18),(6,17),(7,19),(8,21),(9,20),(10,22),(11,24),(12,23)], [(1,10),(2,11),(3,12),(4,7),(5,8),(6,9),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,4),(2,5),(3,6),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(4,7,10),(5,8,11),(6,9,12),(16,19,22),(17,20,23),(18,21,24)]])
G:=TransitiveGroup(24,250);
C2xS3xA4 is a maximal subgroup of
D6:S4 A4:D12
C2xS3xA4 is a maximal quotient of SL2(F3).11D6 Dic6.A4 D12.A4
Matrix representation of C2xS3xA4 ►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 |
| 0 | -1 | 0 | 0 | 0 |
| 1 | -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 | 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 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
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],[0,1,0,0,0,-1,-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,-1,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
C2xS3xA4 in GAP, Magma, Sage, TeX
C_2\times S_3\times A_4
% in TeX
G:=Group("C2xS3xA4"); // GroupNames label
G:=SmallGroup(144,190);
// by ID
G=gap.SmallGroup(144,190);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-3,231,106,3461]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^3=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations
Export