non-abelian, soluble, monomial
Aliases: (C4xC12):1S3, C3:(C42:S3), C42:(C3:S3), C42:C3:2S3, (C2xC6).3S4, C22.(C3:S4), (C3xC42:C3):4C2, SmallGroup(288,401)
Series: Derived ►Chief ►Lower central ►Upper central
| C3xC42:C3 — (C4xC12):S3 |
Generators and relations for (C4xC12):S3
G = < a,b,c,d | a4=b12=c3=d2=1, ab=ba, cac-1=dad=b9, cbc-1=a-1b7, dbd=ab8, dcd=c-1 >
Subgroups: 552 in 60 conjugacy classes, 11 normal (8 characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C8, C2xC4, D4, Q8, C32, Dic3, C12, A4, D6, C2xC6, C42, M4(2), C4oD4, C3:S3, C3:C8, Dic6, C4xS3, D12, C3:D4, C2xC12, S4, C4wrC2, C3xA4, C42:C3, C4.Dic3, C4xC12, C4oD12, C3:S4, C42:4S3, C42:S3, C3xC42:C3, (C4xC12):S3
Quotients: C1, C2, S3, C3:S3, S4, C3:S4, C42:S3, (C4xC12):S3
Character table of (C4xC12):S3
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 12A | 12B | 12C | 12D | |
| size | 1 | 3 | 36 | 2 | 32 | 32 | 32 | 3 | 3 | 6 | 36 | 6 | 36 | 36 | 6 | 6 | 6 | 6 | |
| 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 | 2 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ5 | 2 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ7 | 3 | 3 | -1 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 3 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | 1 | 3 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ9 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | 1 | -1 | -i | i | -1+2i | -1-2i | 1 | 1 | complex lifted from C42:S3 |
| ρ10 | 3 | -1 | -1 | 3 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | 1 | -1 | i | -i | -1-2i | -1+2i | 1 | 1 | complex lifted from C42:S3 |
| ρ11 | 3 | -1 | 1 | 3 | 0 | 0 | 0 | -1-2i | -1+2i | 1 | -1 | -1 | i | -i | -1+2i | -1-2i | 1 | 1 | complex lifted from C42:S3 |
| ρ12 | 3 | -1 | 1 | 3 | 0 | 0 | 0 | -1+2i | -1-2i | 1 | -1 | -1 | -i | i | -1-2i | -1+2i | 1 | 1 | complex lifted from C42:S3 |
| ρ13 | 6 | -2 | 0 | 6 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 2 | 2 | -2 | -2 | orthogonal lifted from C42:S3 |
| ρ14 | 6 | 6 | 0 | -3 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | -3 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from C3:S4 |
| ρ15 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 1 | 0 | 0 | -1 | -1 | 1+2√3 | 1-2√3 | orthogonal faithful |
| ρ16 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 1 | 0 | 0 | -1 | -1 | 1-2√3 | 1+2√3 | orthogonal faithful |
| ρ17 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | -2+4i | -2-4i | 2 | 0 | 1 | 0 | 0 | 1+2i | 1-2i | -1 | -1 | complex faithful |
| ρ18 | 6 | -2 | 0 | -3 | 0 | 0 | 0 | -2-4i | -2+4i | 2 | 0 | 1 | 0 | 0 | 1-2i | 1+2i | -1 | -1 | complex faithful |
►On 36 points
Generators in S36
(1 7 10 6)(2 8 11 4)(3 9 12 5)(25 28 31 34)(26 29 32 35)(27 30 33 36)
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(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 21 30)(2 13 34)(3 17 26)(4 22 25)(5 14 29)(6 18 33)(7 24 27)(8 16 31)(9 20 35)(10 15 36)(11 19 28)(12 23 32)
(1 20)(2 16)(3 24)(4 19)(5 15)(6 23)(7 17)(8 13)(9 21)(10 14)(11 22)(12 18)(25 28)(26 27)(29 36)(30 35)(31 34)(32 33)
G:=sub<Sym(36)| (1,7,10,6)(2,8,11,4)(3,9,12,5)(25,28,31,34)(26,29,32,35)(27,30,33,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,21,30)(2,13,34)(3,17,26)(4,22,25)(5,14,29)(6,18,33)(7,24,27)(8,16,31)(9,20,35)(10,15,36)(11,19,28)(12,23,32), (1,20)(2,16)(3,24)(4,19)(5,15)(6,23)(7,17)(8,13)(9,21)(10,14)(11,22)(12,18)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33)>;
G:=Group( (1,7,10,6)(2,8,11,4)(3,9,12,5)(25,28,31,34)(26,29,32,35)(27,30,33,36), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,21,30)(2,13,34)(3,17,26)(4,22,25)(5,14,29)(6,18,33)(7,24,27)(8,16,31)(9,20,35)(10,15,36)(11,19,28)(12,23,32), (1,20)(2,16)(3,24)(4,19)(5,15)(6,23)(7,17)(8,13)(9,21)(10,14)(11,22)(12,18)(25,28)(26,27)(29,36)(30,35)(31,34)(32,33) );
G=PermutationGroup([[(1,7,10,6),(2,8,11,4),(3,9,12,5),(25,28,31,34),(26,29,32,35),(27,30,33,36)], [(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)], [(1,21,30),(2,13,34),(3,17,26),(4,22,25),(5,14,29),(6,18,33),(7,24,27),(8,16,31),(9,20,35),(10,15,36),(11,19,28),(12,23,32)], [(1,20),(2,16),(3,24),(4,19),(5,15),(6,23),(7,17),(8,13),(9,21),(10,14),(11,22),(12,18),(25,28),(26,27),(29,36),(30,35),(31,34),(32,33)]])
Matrix representation of (C4xC12):S3 ►in GL5(F73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 46 |
| 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 46 |
| 72 | 1 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 27 |
| 0 | 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 47 |
| 0 | 0 | 72 | 72 | 47 |
| 0 | 0 | 0 | 45 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 72 | 47 |
| 0 | 0 | 45 | 0 | 1 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,46,0,0,0,46,0,46],[72,72,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,46,0,0,0,27,27,72],[1,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,72,72,45,0,0,47,47,1],[0,1,0,0,0,1,0,0,0,0,0,0,72,72,45,0,0,0,72,0,0,0,0,47,1] >;
(C4xC12):S3 in GAP, Magma, Sage, TeX
(C_4\times C_{12})\rtimes S_3 % in TeX
G:=Group("(C4xC12):S3"); // GroupNames label
G:=SmallGroup(288,401);
// by ID
G=gap.SmallGroup(288,401);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,57,254,1011,514,360,634,3476,102,9077,3036,5305]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^12=c^3=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=b^9,c*b*c^-1=a^-1*b^7,d*b*d=a*b^8,d*c*d=c^-1>;
// generators/relations
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