direct product, metabelian, soluble, monomial, A-group
Aliases: S3×C42⋊C3, (C4×C12)⋊4C6, (S3×C42)⋊C3, C42⋊4(C3×S3), C22.3(S3×A4), (C22×S3).3A4, C3⋊(C2×C42⋊C3), (C3×C42⋊C3)⋊5C2, (C2×C6).3(C2×A4), SmallGroup(288,407)
Series: Derived ►Chief ►Lower central ►Upper central
| C4×C12 — S3×C42⋊C3 |
Generators and relations for S3×C42⋊C3
G = < a,b,c,d,e | a3=b2=c4=d4=e3=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=cd-1, ede-1=c-1d2 >
Subgroups: 354 in 59 conjugacy classes, 12 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, S3, C6, C2×C4, C23, C32, Dic3, C12, A4, D6, C2×C6, C42, C42, C22×C4, C3×S3, C4×S3, C2×Dic3, C2×C12, C2×A4, C22×S3, C2×C42, C3×A4, C42⋊C3, C42⋊C3, C4×Dic3, C4×C12, S3×C2×C4, S3×A4, C2×C42⋊C3, S3×C42, C3×C42⋊C3, S3×C42⋊C3
Quotients: C1, C2, C3, S3, C6, A4, C3×S3, C2×A4, C42⋊C3, S3×A4, C2×C42⋊C3, S3×C42⋊C3
Character table of S3×C42⋊C3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 3 | 3 | 9 | 2 | 16 | 16 | 32 | 32 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 6 | 48 | 48 | 6 | 6 | 6 | 6 | |
| 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 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 2 | 0 | 0 | -1 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 2 | 0 | 0 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ10 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ11 | 3 | 3 | -3 | -3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from C2×A4 |
| ρ12 | 3 | -1 | 3 | -1 | 3 | 0 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | 1 | -1+2i | -1-2i | 1 | 1 | -1 | 0 | 0 | 1 | 1 | -1+2i | -1-2i | complex lifted from C42⋊C3 |
| ρ13 | 3 | -1 | 3 | -1 | 3 | 0 | 0 | 0 | 0 | 1 | 1 | -1+2i | -1-2i | 1 | 1 | -1+2i | -1-2i | -1 | 0 | 0 | -1+2i | -1-2i | 1 | 1 | complex lifted from C42⋊C3 |
| ρ14 | 3 | -1 | 3 | -1 | 3 | 0 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | 1 | -1-2i | -1+2i | 1 | 1 | -1 | 0 | 0 | 1 | 1 | -1-2i | -1+2i | complex lifted from C42⋊C3 |
| ρ15 | 3 | -1 | 3 | -1 | 3 | 0 | 0 | 0 | 0 | 1 | 1 | -1-2i | -1+2i | 1 | 1 | -1-2i | -1+2i | -1 | 0 | 0 | -1-2i | -1+2i | 1 | 1 | complex lifted from C42⋊C3 |
| ρ16 | 3 | -1 | -3 | 1 | 3 | 0 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | 1 | 1+2i | 1-2i | -1 | -1 | -1 | 0 | 0 | 1 | 1 | -1-2i | -1+2i | complex lifted from C2×C42⋊C3 |
| ρ17 | 3 | -1 | -3 | 1 | 3 | 0 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | 1 | 1-2i | 1+2i | -1 | -1 | -1 | 0 | 0 | 1 | 1 | -1+2i | -1-2i | complex lifted from C2×C42⋊C3 |
| ρ18 | 3 | -1 | -3 | 1 | 3 | 0 | 0 | 0 | 0 | 1 | 1 | -1+2i | -1-2i | -1 | -1 | 1-2i | 1+2i | -1 | 0 | 0 | -1+2i | -1-2i | 1 | 1 | complex lifted from C2×C42⋊C3 |
| ρ19 | 3 | -1 | -3 | 1 | 3 | 0 | 0 | 0 | 0 | 1 | 1 | -1-2i | -1+2i | -1 | -1 | 1+2i | 1-2i | -1 | 0 | 0 | -1-2i | -1+2i | 1 | 1 | complex lifted from C2×C42⋊C3 |
| ρ20 | 6 | 6 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | -3 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from S3×A4 |
| ρ21 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2+4i | -2-4i | 2 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | 1-2i | 1+2i | complex faithful |
| ρ22 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 2 | 2 | -2+4i | -2-4i | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1-2i | 1+2i | -1 | -1 | complex faithful |
| ρ23 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | 2 | 2 | -2-4i | -2+4i | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1+2i | 1-2i | -1 | -1 | complex faithful |
| ρ24 | 6 | -2 | 0 | 0 | -3 | 0 | 0 | 0 | 0 | -2-4i | -2+4i | 2 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | -1 | -1 | 1+2i | 1-2i | complex faithful |
►On 36 points
Generators in S36
(1 6 8)(2 7 11)(3 12 10)(4 5 9)(13 33 28)(14 34 25)(15 35 26)(16 36 27)(17 21 30)(18 22 31)(19 23 32)(20 24 29)
(1 6)(3 10)(4 9)(7 11)(13 33)(14 34)(15 35)(16 36)(17 30)(18 31)(19 32)(20 29)
(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)
(1 9 7 10)(2 12 8 5)(3 6 4 11)(13 14 15 16)(17 19)(18 20)(21 23)(22 24)(25 26 27 28)(29 31)(30 32)(33 34 35 36)
(1 29 14)(2 22 27)(3 17 33)(4 19 35)(5 23 26)(6 20 34)(7 31 16)(8 24 25)(9 32 15)(10 30 13)(11 18 36)(12 21 28)
G:=sub<Sym(36)| (1,6,8)(2,7,11)(3,12,10)(4,5,9)(13,33,28)(14,34,25)(15,35,26)(16,36,27)(17,21,30)(18,22,31)(19,23,32)(20,24,29), (1,6)(3,10)(4,9)(7,11)(13,33)(14,34)(15,35)(16,36)(17,30)(18,31)(19,32)(20,29), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,9,7,10)(2,12,8,5)(3,6,4,11)(13,14,15,16)(17,19)(18,20)(21,23)(22,24)(25,26,27,28)(29,31)(30,32)(33,34,35,36), (1,29,14)(2,22,27)(3,17,33)(4,19,35)(5,23,26)(6,20,34)(7,31,16)(8,24,25)(9,32,15)(10,30,13)(11,18,36)(12,21,28)>;
G:=Group( (1,6,8)(2,7,11)(3,12,10)(4,5,9)(13,33,28)(14,34,25)(15,35,26)(16,36,27)(17,21,30)(18,22,31)(19,23,32)(20,24,29), (1,6)(3,10)(4,9)(7,11)(13,33)(14,34)(15,35)(16,36)(17,30)(18,31)(19,32)(20,29), (13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,9,7,10)(2,12,8,5)(3,6,4,11)(13,14,15,16)(17,19)(18,20)(21,23)(22,24)(25,26,27,28)(29,31)(30,32)(33,34,35,36), (1,29,14)(2,22,27)(3,17,33)(4,19,35)(5,23,26)(6,20,34)(7,31,16)(8,24,25)(9,32,15)(10,30,13)(11,18,36)(12,21,28) );
G=PermutationGroup([[(1,6,8),(2,7,11),(3,12,10),(4,5,9),(13,33,28),(14,34,25),(15,35,26),(16,36,27),(17,21,30),(18,22,31),(19,23,32),(20,24,29)], [(1,6),(3,10),(4,9),(7,11),(13,33),(14,34),(15,35),(16,36),(17,30),(18,31),(19,32),(20,29)], [(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(1,9,7,10),(2,12,8,5),(3,6,4,11),(13,14,15,16),(17,19),(18,20),(21,23),(22,24),(25,26,27,28),(29,31),(30,32),(33,34,35,36)], [(1,29,14),(2,22,27),(3,17,33),(4,19,35),(5,23,26),(6,20,34),(7,31,16),(8,24,25),(9,32,15),(10,30,13),(11,18,36),(12,21,28)]])
Matrix representation of S3×C42⋊C3 ►in GL5(𝔽13)
| 0 | 12 | 0 | 0 | 0 |
| 1 | 12 | 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 | 5 | 2 | 5 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 5 | 9 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 11 | 10 | 4 |
G:=sub<GL(5,GF(13))| [0,1,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[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],[1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,2,8,0,0,0,5,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,9,12,0,0,0,0,0,5],[1,0,0,0,0,0,1,0,0,0,0,0,9,0,11,0,0,0,0,10,0,0,0,1,4] >;
S3×C42⋊C3 in GAP, Magma, Sage, TeX
S_3\times C_4^2\rtimes C_3
% in TeX
G:=Group("S3xC4^2:C3"); // GroupNames label
G:=SmallGroup(288,407);
// by ID
G=gap.SmallGroup(288,407);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-3,-2,2,198,772,2110,360,1684,3036,5305]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^4=d^4=e^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^-1,e*d*e^-1=c^-1*d^2>;
// generators/relations
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