direct product, metabelian, supersoluble, monomial
Aliases: C6×Dic12, C24.83D6, C12.69D12, C62.88D4, C6⋊1(C3×Q16), C3⋊1(C6×Q16), (C3×C6)⋊4Q16, C8.16(S3×C6), C6.12(C6×D4), C4.8(C3×D12), C32⋊9(C2×Q16), (C2×C24).10C6, C24.20(C2×C6), (C2×C24).21S3, (C6×C24).12C2, C12.31(C3×D4), (C2×C6).75D12, C2.14(C6×D12), C6.100(C2×D12), (C2×C12).444D6, (C3×C12).133D4, (C2×Dic6).4C6, Dic6.7(C2×C6), C12.31(C22×C6), (C3×C24).58C22, (C6×Dic6).19C2, C22.14(C3×D12), C12.218(C22×S3), (C3×C12).163C23, (C6×C12).323C22, (C3×Dic6).47C22, C4.29(S3×C2×C6), (C2×C8).4(C3×S3), (C2×C4).81(S3×C6), (C2×C6).23(C3×D4), (C3×C6).182(C2×D4), (C2×C12).108(C2×C6), SmallGroup(288,676)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×Dic12
G = < a,b,c | a6=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 282 in 131 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C8, Q16, C2×Q8, C3×C6, C3×C6, C24, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C2×Q16, C3×Dic3, C3×C12, C62, Dic12, C2×C24, C2×C24, C3×Q16, C2×Dic6, C6×Q8, C3×C24, C3×Dic6, C3×Dic6, C6×Dic3, C6×C12, C2×Dic12, C6×Q16, C3×Dic12, C6×C24, C6×Dic6, C6×Dic12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, Q16, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C2×Q16, S3×C6, Dic12, C3×Q16, C2×D12, C6×D4, C3×D12, S3×C2×C6, C2×Dic12, C6×Q16, C3×Dic12, C6×D12, C6×Dic12
►On 96 points
(1 66 9 50 17 58)(2 67 10 51 18 59)(3 68 11 52 19 60)(4 69 12 53 20 61)(5 70 13 54 21 62)(6 71 14 55 22 63)(7 72 15 56 23 64)(8 49 16 57 24 65)(25 95 41 87 33 79)(26 96 42 88 34 80)(27 73 43 89 35 81)(28 74 44 90 36 82)(29 75 45 91 37 83)(30 76 46 92 38 84)(31 77 47 93 39 85)(32 78 48 94 40 86)
(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)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)
(1 95 13 83)(2 94 14 82)(3 93 15 81)(4 92 16 80)(5 91 17 79)(6 90 18 78)(7 89 19 77)(8 88 20 76)(9 87 21 75)(10 86 22 74)(11 85 23 73)(12 84 24 96)(25 70 37 58)(26 69 38 57)(27 68 39 56)(28 67 40 55)(29 66 41 54)(30 65 42 53)(31 64 43 52)(32 63 44 51)(33 62 45 50)(34 61 46 49)(35 60 47 72)(36 59 48 71)
G:=sub<Sym(96)| (1,66,9,50,17,58)(2,67,10,51,18,59)(3,68,11,52,19,60)(4,69,12,53,20,61)(5,70,13,54,21,62)(6,71,14,55,22,63)(7,72,15,56,23,64)(8,49,16,57,24,65)(25,95,41,87,33,79)(26,96,42,88,34,80)(27,73,43,89,35,81)(28,74,44,90,36,82)(29,75,45,91,37,83)(30,76,46,92,38,84)(31,77,47,93,39,85)(32,78,48,94,40,86), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,95,13,83)(2,94,14,82)(3,93,15,81)(4,92,16,80)(5,91,17,79)(6,90,18,78)(7,89,19,77)(8,88,20,76)(9,87,21,75)(10,86,22,74)(11,85,23,73)(12,84,24,96)(25,70,37,58)(26,69,38,57)(27,68,39,56)(28,67,40,55)(29,66,41,54)(30,65,42,53)(31,64,43,52)(32,63,44,51)(33,62,45,50)(34,61,46,49)(35,60,47,72)(36,59,48,71)>;
G:=Group( (1,66,9,50,17,58)(2,67,10,51,18,59)(3,68,11,52,19,60)(4,69,12,53,20,61)(5,70,13,54,21,62)(6,71,14,55,22,63)(7,72,15,56,23,64)(8,49,16,57,24,65)(25,95,41,87,33,79)(26,96,42,88,34,80)(27,73,43,89,35,81)(28,74,44,90,36,82)(29,75,45,91,37,83)(30,76,46,92,38,84)(31,77,47,93,39,85)(32,78,48,94,40,86), (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)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96), (1,95,13,83)(2,94,14,82)(3,93,15,81)(4,92,16,80)(5,91,17,79)(6,90,18,78)(7,89,19,77)(8,88,20,76)(9,87,21,75)(10,86,22,74)(11,85,23,73)(12,84,24,96)(25,70,37,58)(26,69,38,57)(27,68,39,56)(28,67,40,55)(29,66,41,54)(30,65,42,53)(31,64,43,52)(32,63,44,51)(33,62,45,50)(34,61,46,49)(35,60,47,72)(36,59,48,71) );
G=PermutationGroup([[(1,66,9,50,17,58),(2,67,10,51,18,59),(3,68,11,52,19,60),(4,69,12,53,20,61),(5,70,13,54,21,62),(6,71,14,55,22,63),(7,72,15,56,23,64),(8,49,16,57,24,65),(25,95,41,87,33,79),(26,96,42,88,34,80),(27,73,43,89,35,81),(28,74,44,90,36,82),(29,75,45,91,37,83),(30,76,46,92,38,84),(31,77,47,93,39,85),(32,78,48,94,40,86)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)], [(1,95,13,83),(2,94,14,82),(3,93,15,81),(4,92,16,80),(5,91,17,79),(6,90,18,78),(7,89,19,77),(8,88,20,76),(9,87,21,75),(10,86,22,74),(11,85,23,73),(12,84,24,96),(25,70,37,58),(26,69,38,57),(27,68,39,56),(28,67,40,55),(29,66,41,54),(30,65,42,53),(31,64,43,52),(32,63,44,51),(33,62,45,50),(34,61,46,49),(35,60,47,72),(36,59,48,71)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12X | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | Q16 | C3×S3 | D12 | C3×D4 | D12 | C3×D4 | S3×C6 | S3×C6 | Dic12 | C3×Q16 | C3×D12 | C3×D12 | C3×Dic12 |
| kernel | C6×Dic12 | C3×Dic12 | C6×C24 | C6×Dic6 | C2×Dic12 | Dic12 | C2×C24 | C2×Dic6 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C3×C6 | C2×C8 | C12 | C12 | C2×C6 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 16 |
Matrix representation of C6×Dic12 ►in GL4(𝔽73) generated by
| 65 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 65 |
| 70 | 0 | 0 | 0 |
| 56 | 24 | 0 | 0 |
| 0 | 0 | 66 | 0 |
| 0 | 0 | 0 | 52 |
| 46 | 30 | 0 | 0 |
| 39 | 27 | 0 | 0 |
| 0 | 0 | 0 | 10 |
| 0 | 0 | 51 | 0 |
G:=sub<GL(4,GF(73))| [65,0,0,0,0,65,0,0,0,0,65,0,0,0,0,65],[70,56,0,0,0,24,0,0,0,0,66,0,0,0,0,52],[46,39,0,0,30,27,0,0,0,0,0,51,0,0,10,0] >;
C6×Dic12 in GAP, Magma, Sage, TeX
C_6\times {\rm Dic}_{12} % in TeX
G:=Group("C6xDic12"); // GroupNames label
G:=SmallGroup(288,676);
// by ID
G=gap.SmallGroup(288,676);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,590,394,2524,102,9414]);
// Polycyclic
G:=Group<a,b,c|a^6=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations