direct product, metabelian, supersoluble, monomial
Aliases: C3×C12⋊2Q8, C12⋊8Dic6, C12.65D12, C122.8C2, C62.160C23, (C3×C12)⋊9Q8, C12⋊2(C3×Q8), C6.1(C6×D4), C6.2(C6×Q8), (C4×C12).8C6, C4⋊2(C3×Dic6), C4.4(C3×D12), C2.4(C6×D12), C32⋊9(C4⋊Q8), (C4×C12).24S3, C12.27(C3×D4), C6.89(C2×D12), C4⋊Dic3.4C6, C42.4(C3×S3), C2.4(C6×Dic6), (C2×C12).437D6, (C3×C12).129D4, (C2×Dic6).2C6, C6.48(C2×Dic6), (C6×Dic6).17C2, (C6×C12).319C22, (C6×Dic3).87C22, C3⋊1(C3×C4⋊Q8), (C2×C4).75(S3×C6), C22.34(S3×C2×C6), (C3×C6).45(C2×Q8), (C3×C6).172(C2×D4), (C2×C12).104(C2×C6), (C3×C4⋊Dic3).23C2, (C2×C6).15(C22×C6), (C2×Dic3).1(C2×C6), (C2×C6).293(C22×S3), SmallGroup(288,640)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12⋊2Q8
G = < a,b,c,d | a3=b12=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 306 in 151 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C4⋊C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C4⋊Q8, C3×Dic3, C3×C12, C62, C4⋊Dic3, C4×C12, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, C3×Dic6, C6×Dic3, C6×C12, C6×C12, C12⋊2Q8, C3×C4⋊Q8, C3×C4⋊Dic3, C122, C6×Dic6, C3×C12⋊2Q8
Quotients: C1, C2, C3, C22, S3, C6, D4, Q8, C23, D6, C2×C6, C2×D4, C2×Q8, C3×S3, Dic6, D12, C3×D4, C3×Q8, C22×S3, C22×C6, C4⋊Q8, S3×C6, C2×Dic6, C2×D12, C6×D4, C6×Q8, C3×Dic6, C3×D12, S3×C2×C6, C12⋊2Q8, C3×C4⋊Q8, C6×Dic6, C6×D12, C3×C12⋊2Q8
►On 96 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 89 93)(86 90 94)(87 91 95)(88 92 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 81 21 87)(2 82 22 88)(3 83 23 89)(4 84 24 90)(5 73 13 91)(6 74 14 92)(7 75 15 93)(8 76 16 94)(9 77 17 95)(10 78 18 96)(11 79 19 85)(12 80 20 86)(25 67 44 56)(26 68 45 57)(27 69 46 58)(28 70 47 59)(29 71 48 60)(30 72 37 49)(31 61 38 50)(32 62 39 51)(33 63 40 52)(34 64 41 53)(35 65 42 54)(36 66 43 55)
(1 36 21 43)(2 35 22 42)(3 34 23 41)(4 33 24 40)(5 32 13 39)(6 31 14 38)(7 30 15 37)(8 29 16 48)(9 28 17 47)(10 27 18 46)(11 26 19 45)(12 25 20 44)(49 93 72 75)(50 92 61 74)(51 91 62 73)(52 90 63 84)(53 89 64 83)(54 88 65 82)(55 87 66 81)(56 86 67 80)(57 85 68 79)(58 96 69 78)(59 95 70 77)(60 94 71 76)
G:=sub<Sym(96)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (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), (1,81,21,87)(2,82,22,88)(3,83,23,89)(4,84,24,90)(5,73,13,91)(6,74,14,92)(7,75,15,93)(8,76,16,94)(9,77,17,95)(10,78,18,96)(11,79,19,85)(12,80,20,86)(25,67,44,56)(26,68,45,57)(27,69,46,58)(28,70,47,59)(29,71,48,60)(30,72,37,49)(31,61,38,50)(32,62,39,51)(33,63,40,52)(34,64,41,53)(35,65,42,54)(36,66,43,55), (1,36,21,43)(2,35,22,42)(3,34,23,41)(4,33,24,40)(5,32,13,39)(6,31,14,38)(7,30,15,37)(8,29,16,48)(9,28,17,47)(10,27,18,46)(11,26,19,45)(12,25,20,44)(49,93,72,75)(50,92,61,74)(51,91,62,73)(52,90,63,84)(53,89,64,83)(54,88,65,82)(55,87,66,81)(56,86,67,80)(57,85,68,79)(58,96,69,78)(59,95,70,77)(60,94,71,76)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (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), (1,81,21,87)(2,82,22,88)(3,83,23,89)(4,84,24,90)(5,73,13,91)(6,74,14,92)(7,75,15,93)(8,76,16,94)(9,77,17,95)(10,78,18,96)(11,79,19,85)(12,80,20,86)(25,67,44,56)(26,68,45,57)(27,69,46,58)(28,70,47,59)(29,71,48,60)(30,72,37,49)(31,61,38,50)(32,62,39,51)(33,63,40,52)(34,64,41,53)(35,65,42,54)(36,66,43,55), (1,36,21,43)(2,35,22,42)(3,34,23,41)(4,33,24,40)(5,32,13,39)(6,31,14,38)(7,30,15,37)(8,29,16,48)(9,28,17,47)(10,27,18,46)(11,26,19,45)(12,25,20,44)(49,93,72,75)(50,92,61,74)(51,91,62,73)(52,90,63,84)(53,89,64,83)(54,88,65,82)(55,87,66,81)(56,86,67,80)(57,85,68,79)(58,96,69,78)(59,95,70,77)(60,94,71,76) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,89,93),(86,90,94),(87,91,95),(88,92,96)], [(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)], [(1,81,21,87),(2,82,22,88),(3,83,23,89),(4,84,24,90),(5,73,13,91),(6,74,14,92),(7,75,15,93),(8,76,16,94),(9,77,17,95),(10,78,18,96),(11,79,19,85),(12,80,20,86),(25,67,44,56),(26,68,45,57),(27,69,46,58),(28,70,47,59),(29,71,48,60),(30,72,37,49),(31,61,38,50),(32,62,39,51),(33,63,40,52),(34,64,41,53),(35,65,42,54),(36,66,43,55)], [(1,36,21,43),(2,35,22,42),(3,34,23,41),(4,33,24,40),(5,32,13,39),(6,31,14,38),(7,30,15,37),(8,29,16,48),(9,28,17,47),(10,27,18,46),(11,26,19,45),(12,25,20,44),(49,93,72,75),(50,92,61,74),(51,91,62,73),(52,90,63,84),(53,89,64,83),(54,88,65,82),(55,87,66,81),(56,86,67,80),(57,85,68,79),(58,96,69,78),(59,95,70,77),(60,94,71,76)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | ··· | 6O | 12A | ··· | 12AV | 12AW | ··· | 12BD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | Q8 | D6 | C3×S3 | Dic6 | D12 | C3×D4 | C3×Q8 | S3×C6 | C3×Dic6 | C3×D12 |
| kernel | C3×C12⋊2Q8 | C3×C4⋊Dic3 | C122 | C6×Dic6 | C12⋊2Q8 | C4⋊Dic3 | C4×C12 | C2×Dic6 | C4×C12 | C3×C12 | C3×C12 | C2×C12 | C42 | C12 | C12 | C12 | C12 | C2×C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 1 | 2 | 4 | 3 | 2 | 8 | 4 | 4 | 8 | 6 | 16 | 8 |
Matrix representation of C3×C12⋊2Q8 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 11 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 6 |
| 8 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[11,0,0,0,0,6,0,0,0,0,11,0,0,0,0,6],[8,0,0,0,0,5,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C12⋊2Q8 in GAP, Magma, Sage, TeX
C_3\times C_{12}\rtimes_2Q_8 % in TeX
G:=Group("C3xC12:2Q8"); // GroupNames label
G:=SmallGroup(288,640);
// by ID
G=gap.SmallGroup(288,640);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,344,590,142,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations