direct product, metabelian, supersoluble, monomial
Aliases: C12×Dic6, C122.7C2, C62.159C23, C3⋊1(Q8×C12), C12⋊3(C3×Q8), C6.1(C6×Q8), C32⋊9(C4×Q8), C4.9(S3×C12), (C3×C12)⋊10Q8, C12.87(C4×S3), (C4×C12).11C6, (C4×C12).23S3, C42.3(C3×S3), C2.1(C6×Dic6), C12.19(C2×C12), (C2×C12).436D6, Dic3⋊C4.7C6, C4⋊Dic3.13C6, C6.1(C22×C12), (C4×Dic3).7C6, C6.47(C2×Dic6), (C6×Dic6).20C2, (C2×Dic6).10C6, Dic3.1(C2×C12), C6.110(C4○D12), (C6×C12).278C22, (Dic3×C12).18C2, (C6×Dic3).157C22, C2.4(S3×C2×C12), C6.100(S3×C2×C4), C6.1(C3×C4○D4), C22.8(S3×C2×C6), (C2×C4).95(S3×C6), C2.1(C3×C4○D12), (C3×C6).44(C2×Q8), (C3×C12).99(C2×C4), (C2×C12).125(C2×C6), (C3×C6).91(C4○D4), (C3×C4⋊Dic3).27C2, (C3×C6).72(C22×C4), (C2×C6).14(C22×C6), (C3×Dic3⋊C4).16C2, (C2×C6).292(C22×S3), (C3×Dic3).17(C2×C4), (C2×Dic3).15(C2×C6), SmallGroup(288,639)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×Dic6
G = < a,b,c | a12=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 274 in 155 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C32, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×Q8, C3×Dic3, C3×Dic3, C3×C12, C3×C12, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, C3×Dic6, C6×Dic3, C6×C12, C4×Dic6, Q8×C12, Dic3×C12, C3×Dic3⋊C4, C3×C4⋊Dic3, C122, C6×Dic6, C12×Dic6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Q8, C23, C12, D6, C2×C6, C22×C4, C2×Q8, C4○D4, C3×S3, Dic6, C4×S3, C2×C12, C3×Q8, C22×S3, C22×C6, C4×Q8, S3×C6, C2×Dic6, S3×C2×C4, C4○D12, C22×C12, C6×Q8, C3×C4○D4, C3×Dic6, S3×C12, S3×C2×C6, C4×Dic6, Q8×C12, C6×Dic6, S3×C2×C12, C3×C4○D12, C12×Dic6
►On 96 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)(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 89 42 76 9 85 38 84 5 93 46 80)(2 90 43 77 10 86 39 73 6 94 47 81)(3 91 44 78 11 87 40 74 7 95 48 82)(4 92 45 79 12 88 41 75 8 96 37 83)(13 25 55 72 17 29 59 64 21 33 51 68)(14 26 56 61 18 30 60 65 22 34 52 69)(15 27 57 62 19 31 49 66 23 35 53 70)(16 28 58 63 20 32 50 67 24 36 54 71)
(1 16 38 50)(2 17 39 51)(3 18 40 52)(4 19 41 53)(5 20 42 54)(6 21 43 55)(7 22 44 56)(8 23 45 57)(9 24 46 58)(10 13 47 59)(11 14 48 60)(12 15 37 49)(25 94 64 77)(26 95 65 78)(27 96 66 79)(28 85 67 80)(29 86 68 81)(30 87 69 82)(31 88 70 83)(32 89 71 84)(33 90 72 73)(34 91 61 74)(35 92 62 75)(36 93 63 76)
G:=sub<Sym(96)| (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), (1,89,42,76,9,85,38,84,5,93,46,80)(2,90,43,77,10,86,39,73,6,94,47,81)(3,91,44,78,11,87,40,74,7,95,48,82)(4,92,45,79,12,88,41,75,8,96,37,83)(13,25,55,72,17,29,59,64,21,33,51,68)(14,26,56,61,18,30,60,65,22,34,52,69)(15,27,57,62,19,31,49,66,23,35,53,70)(16,28,58,63,20,32,50,67,24,36,54,71), (1,16,38,50)(2,17,39,51)(3,18,40,52)(4,19,41,53)(5,20,42,54)(6,21,43,55)(7,22,44,56)(8,23,45,57)(9,24,46,58)(10,13,47,59)(11,14,48,60)(12,15,37,49)(25,94,64,77)(26,95,65,78)(27,96,66,79)(28,85,67,80)(29,86,68,81)(30,87,69,82)(31,88,70,83)(32,89,71,84)(33,90,72,73)(34,91,61,74)(35,92,62,75)(36,93,63,76)>;
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)(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), (1,89,42,76,9,85,38,84,5,93,46,80)(2,90,43,77,10,86,39,73,6,94,47,81)(3,91,44,78,11,87,40,74,7,95,48,82)(4,92,45,79,12,88,41,75,8,96,37,83)(13,25,55,72,17,29,59,64,21,33,51,68)(14,26,56,61,18,30,60,65,22,34,52,69)(15,27,57,62,19,31,49,66,23,35,53,70)(16,28,58,63,20,32,50,67,24,36,54,71), (1,16,38,50)(2,17,39,51)(3,18,40,52)(4,19,41,53)(5,20,42,54)(6,21,43,55)(7,22,44,56)(8,23,45,57)(9,24,46,58)(10,13,47,59)(11,14,48,60)(12,15,37,49)(25,94,64,77)(26,95,65,78)(27,96,66,79)(28,85,67,80)(29,86,68,81)(30,87,69,82)(31,88,70,83)(32,89,71,84)(33,90,72,73)(34,91,61,74)(35,92,62,75)(36,93,63,76) );
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),(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)], [(1,89,42,76,9,85,38,84,5,93,46,80),(2,90,43,77,10,86,39,73,6,94,47,81),(3,91,44,78,11,87,40,74,7,95,48,82),(4,92,45,79,12,88,41,75,8,96,37,83),(13,25,55,72,17,29,59,64,21,33,51,68),(14,26,56,61,18,30,60,65,22,34,52,69),(15,27,57,62,19,31,49,66,23,35,53,70),(16,28,58,63,20,32,50,67,24,36,54,71)], [(1,16,38,50),(2,17,39,51),(3,18,40,52),(4,19,41,53),(5,20,42,54),(6,21,43,55),(7,22,44,56),(8,23,45,57),(9,24,46,58),(10,13,47,59),(11,14,48,60),(12,15,37,49),(25,94,64,77),(26,95,65,78),(27,96,66,79),(28,85,67,80),(29,86,68,81),(30,87,69,82),(31,88,70,83),(32,89,71,84),(33,90,72,73),(34,91,61,74),(35,92,62,75),(36,93,63,76)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 6A | ··· | 6F | 6G | ··· | 6O | 12A | ··· | 12H | 12I | ··· | 12AZ | 12BA | ··· | 12BP |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | + | - | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | S3 | Q8 | D6 | C4○D4 | C3×S3 | Dic6 | C4×S3 | C3×Q8 | S3×C6 | C4○D12 | C3×C4○D4 | C3×Dic6 | S3×C12 | C3×C4○D12 |
| kernel | C12×Dic6 | Dic3×C12 | C3×Dic3⋊C4 | C3×C4⋊Dic3 | C122 | C6×Dic6 | C4×Dic6 | C3×Dic6 | C4×Dic3 | Dic3⋊C4 | C4⋊Dic3 | C4×C12 | C2×Dic6 | Dic6 | C4×C12 | C3×C12 | C2×C12 | C3×C6 | C42 | C12 | C12 | C12 | C2×C4 | C6 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 8 | 4 | 4 | 2 | 2 | 2 | 16 | 1 | 2 | 3 | 2 | 2 | 4 | 4 | 4 | 6 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C12×Dic6 ►in GL3(𝔽13) generated by
| 7 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 1 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 7 |
| 12 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(13))| [7,0,0,0,4,0,0,0,4],[1,0,0,0,2,0,0,0,7],[12,0,0,0,0,12,0,1,0] >;
C12×Dic6 in GAP, Magma, Sage, TeX
C_{12}\times {\rm Dic}_6 % in TeX
G:=Group("C12xDic6"); // GroupNames label
G:=SmallGroup(288,639);
// by ID
G=gap.SmallGroup(288,639);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,344,142,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations