direct product, metabelian, soluble, monomial, A-group
Aliases: C2×S3×A4, C6⋊(C2×A4), C3⋊(C22×A4), (C22×C6)⋊C6, (C6×A4)⋊3C2, (S3×C23)⋊C3, (C22×S3)⋊C6, C22⋊2(S3×C6), C23⋊2(C3×S3), (C3×A4)⋊4C22, (C2×C6)⋊(C2×C6), SmallGroup(144,190)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C6 — C2×S3×A4 |
Generators and relations for C2×S3×A4
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, C2×C6, C2×C6, C24, C3×S3, C3×C6, C2×A4, C2×A4, C22×S3, C22×S3, C22×C6, C3×A4, S3×C6, C22×A4, S3×C23, S3×A4, C6×A4, C2×S3×A4
Quotients: C1, C2, C3, C22, S3, C6, A4, D6, C2×C6, C3×S3, C2×A4, S3×C6, C22×A4, S3×A4, C2×S3×A4
Character table of C2×S3×A4
| 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 S3×C6 |
| 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 S3×C6 |
| 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 C3×S3 |
| 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 C3×S3 |
| 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 C2×A4 |
| 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 C2×A4 |
| 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 C2×A4 |
| 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 S3×A4 |
►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);
C2×S3×A4 is a maximal subgroup of
D6⋊S4 A4⋊D12
C2×S3×A4 is a maximal quotient of SL2(𝔽3).11D6 Dic6.A4 D12.A4
Matrix representation of C2×S3×A4 ►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] >;
C2×S3×A4 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
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