direct product, metabelian, soluble, monomial, A-group
Aliases: S3xC42:C3, (C4xC12):4C6, (S3xC42):C3, C42:4(C3xS3), C22.3(S3xA4), (C22xS3).3A4, C3:(C2xC42:C3), (C3xC42:C3):5C2, (C2xC6).3(C2xA4), SmallGroup(288,407)
Series: Derived ►Chief ►Lower central ►Upper central
| C4xC12 — S3xC42:C3 |
Generators and relations for S3xC42: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, C2xC4, C23, C32, Dic3, C12, A4, D6, C2xC6, C42, C42, C22xC4, C3xS3, C4xS3, C2xDic3, C2xC12, C2xA4, C22xS3, C2xC42, C3xA4, C42:C3, C42:C3, C4xDic3, C4xC12, S3xC2xC4, S3xA4, C2xC42:C3, S3xC42, C3xC42:C3, S3xC42:C3
Quotients: C1, C2, C3, S3, C6, A4, C3xS3, C2xA4, C42:C3, S3xA4, C2xC42:C3, S3xC42:C3
Character table of S3xC42: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 C3xS3 |
| 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 C3xS3 |
| 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 C2xA4 |
| 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 C2xC42: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 C2xC42: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 C2xC42: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 C2xC42: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 S3xA4 |
| 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 S3xC42: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] >;
S3xC42: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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