Aliases: (C6×C12)⋊2C4, (C3×C6).5C42, C62.9(C2×C4), C2.2(C62⋊C4), C32⋊3(C2.C42), (C2×C32⋊C4)⋊3C4, C2.5(C4×C32⋊C4), C3⋊S3.4(C4⋊C4), (C2×C3⋊S3).7Q8, (C2×C4)⋊2(C32⋊C4), (C2×C3⋊Dic3)⋊6C4, (C2×C3⋊S3).50D4, (C3×C6).17(C4⋊C4), C3⋊S3.6(C22⋊C4), C2.3(C4⋊(C32⋊C4)), (C22×C32⋊C4).3C2, C22.13(C2×C32⋊C4), (C3×C6).15(C22⋊C4), (C22×C3⋊S3).62C22, (C2×C4×C3⋊S3).14C2, (C2×C3⋊S3).24(C2×C4), SmallGroup(288,429)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C6×C12)⋊2C4
G = < a,b,c | a6=b12=c4=1, ab=ba, cac-1=a-1b8, cbc-1=a-1b >
Subgroups: 696 in 130 conjugacy classes, 32 normal (18 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C22×C4, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C2.C42, C3⋊Dic3, C3×C12, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, C2×C32⋊C4, C2×C32⋊C4, C22×C3⋊S3, C2×C4×C3⋊S3, C22×C32⋊C4, (C6×C12)⋊2C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C32⋊C4, C2×C32⋊C4, C4×C32⋊C4, C4⋊(C32⋊C4), C62⋊C4, (C6×C12)⋊2C4
►On 48 points
(1 17 5 21 9 14)(2 18 6 22 10 15)(3 19 7 23 11 16)(4 20 8 24 12 13)(25 39 33 47 29 43)(26 40 34 48 30 44)(27 41 35 37 31 45)(28 42 36 38 32 46)
(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)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 38 21 28)(2 25 22 47)(3 44 23 34)(4 31 24 41)(5 42 17 36)(6 29 18 43)(7 48 19 30)(8 35 20 37)(9 46 14 32)(10 33 15 39)(11 40 16 26)(12 27 13 45)
G:=sub<Sym(48)| (1,17,5,21,9,14)(2,18,6,22,10,15)(3,19,7,23,11,16)(4,20,8,24,12,13)(25,39,33,47,29,43)(26,40,34,48,30,44)(27,41,35,37,31,45)(28,42,36,38,32,46), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,38,21,28)(2,25,22,47)(3,44,23,34)(4,31,24,41)(5,42,17,36)(6,29,18,43)(7,48,19,30)(8,35,20,37)(9,46,14,32)(10,33,15,39)(11,40,16,26)(12,27,13,45)>;
G:=Group( (1,17,5,21,9,14)(2,18,6,22,10,15)(3,19,7,23,11,16)(4,20,8,24,12,13)(25,39,33,47,29,43)(26,40,34,48,30,44)(27,41,35,37,31,45)(28,42,36,38,32,46), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,38,21,28)(2,25,22,47)(3,44,23,34)(4,31,24,41)(5,42,17,36)(6,29,18,43)(7,48,19,30)(8,35,20,37)(9,46,14,32)(10,33,15,39)(11,40,16,26)(12,27,13,45) );
G=PermutationGroup([[(1,17,5,21,9,14),(2,18,6,22,10,15),(3,19,7,23,11,16),(4,20,8,24,12,13),(25,39,33,47,29,43),(26,40,34,48,30,44),(27,41,35,37,31,45),(28,42,36,38,32,46)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,38,21,28),(2,25,22,47),(3,44,23,34),(4,31,24,41),(5,42,17,36),(6,29,18,43),(7,48,19,30),(8,35,20,37),(9,46,14,32),(10,33,15,39),(11,40,16,26),(12,27,13,45)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | ··· | 4L | 6A | ··· | 6F | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 4 | 4 | 2 | 2 | 18 | ··· | 18 | 4 | ··· | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | + | |||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | C32⋊C4 | C2×C32⋊C4 | C4×C32⋊C4 | C4⋊(C32⋊C4) | C62⋊C4 |
| kernel | (C6×C12)⋊2C4 | C2×C4×C3⋊S3 | C22×C32⋊C4 | C2×C3⋊Dic3 | C6×C12 | C2×C32⋊C4 | C2×C3⋊S3 | C2×C3⋊S3 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 3 | 1 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of (C6×C12)⋊2C4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 5 | 10 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 4 | 9 | 0 | 0 | 0 | 0 |
| 7 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,1,0,0,0,0,0,0,0,1,12,0,0,0,0,1,0],[5,8,0,0,0,0,10,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,5,0,0,0,0,8,0],[4,7,0,0,0,0,9,9,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
(C6×C12)⋊2C4 in GAP, Magma, Sage, TeX
(C_6\times C_{12})\rtimes_2C_4 % in TeX
G:=Group("(C6xC12):2C4"); // GroupNames label
G:=SmallGroup(288,429);
// by ID
G=gap.SmallGroup(288,429);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,64,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c|a^6=b^12=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^8,c*b*c^-1=a^-1*b>;
// generators/relations