direct product, non-abelian, soluble, monomial
Aliases: C12×S4, (C2×S4).C6, A4⋊C4⋊2C6, (C4×A4)⋊2C6, C2.1(C6×S4), C22⋊(S3×C12), (C12×A4)⋊4C2, A4⋊1(C2×C12), (C6×S4).2C2, C6.40(C2×S4), (C22×C12)⋊1S3, C23.2(S3×C6), (C22×C6).9D6, (C6×A4).9C22, (C2×C6)⋊3(C4×S3), (C3×A4⋊C4)⋊5C2, (C3×A4)⋊4(C2×C4), (C2×A4).2(C2×C6), (C22×C4)⋊1(C3×S3), SmallGroup(288,897)
Series: Derived ►Chief ►Lower central ►Upper central
| A4 — C12×S4 |
Generators and relations for C12×S4
G = < a,b,c,d,e | a12=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, dcd-1=b, ce=ec, ede=d-1 >
Subgroups: 374 in 118 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, C12, A4, A4, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C3×S3, C3×C6, C4×S3, C2×C12, C3×D4, S4, C2×A4, C2×A4, C22×C6, C22×C6, C4×D4, C3×Dic3, C3×C12, C3×A4, S3×C6, C4×C12, C3×C22⋊C4, C3×C4⋊C4, A4⋊C4, C4×A4, C4×A4, C22×C12, C22×C12, C6×D4, C2×S4, S3×C12, C3×S4, C6×A4, D4×C12, C4×S4, C3×A4⋊C4, C12×A4, C6×S4, C12×S4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, C3×S3, C4×S3, C2×C12, S4, S3×C6, C2×S4, S3×C12, C3×S4, C4×S4, C6×S4, C12×S4
►On 36 points
(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 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)
(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)
(1 15 33)(2 16 34)(3 17 35)(4 18 36)(5 19 25)(6 20 26)(7 21 27)(8 22 28)(9 23 29)(10 24 30)(11 13 31)(12 14 32)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)
G:=sub<Sym(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), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,15,33)(2,16,34)(3,17,35)(4,18,36)(5,19,25)(6,20,26)(7,21,27)(8,22,28)(9,23,29)(10,24,30)(11,13,31)(12,14,32), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)>;
G:=Group( (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), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36), (1,15,33)(2,16,34)(3,17,35)(4,18,36)(5,19,25)(6,20,26)(7,21,27)(8,22,28)(9,23,29)(10,24,30)(11,13,31)(12,14,32), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36) );
G=PermutationGroup([[(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)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36)], [(1,15,33),(2,16,34),(3,17,35),(4,18,36),(5,19,25),(6,20,26),(7,21,27),(8,22,28),(9,23,29),(10,24,30),(11,13,31),(12,14,32)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12T | 12U | ··· | 12Z |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 3 | 3 | 6 | 6 | 1 | 1 | 8 | 8 | 8 | 1 | 1 | 3 | 3 | 6 | ··· | 6 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | ··· | 6 | 8 | ··· | 8 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C12 | S4 | C2×S4 | C3×S4 | C4×S4 | C6×S4 | C12×S4 |
| kernel | C12×S4 | C3×A4⋊C4 | C12×A4 | C6×S4 | C4×S4 | C3×S4 | A4⋊C4 | C4×A4 | C2×S4 | S4 | C22×C12 | C22×C6 | C22×C4 | C2×C6 | C23 | C22 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 8 |
Matrix representation of C12×S4 ►in GL3(𝔽13) generated by
| 2 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 12 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(13))| [2,0,0,0,2,0,0,0,2],[12,0,0,0,12,0,0,0,1],[1,0,0,0,12,0,0,0,12],[0,1,0,0,0,1,1,0,0],[12,0,0,0,0,12,0,12,0] >;
C12×S4 in GAP, Magma, Sage, TeX
C_{12}\times S_4 % in TeX
G:=Group("C12xS4"); // GroupNames label
G:=SmallGroup(288,897);
// by ID
G=gap.SmallGroup(288,897);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-3,-2,2,92,1684,6053,285,3534,475]);
// Polycyclic
G:=Group<a,b,c,d,e|a^12=b^2=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,d*c*d^-1=b,c*e=e*c,e*d*e=d^-1>;
// generators/relations