metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊6SD16, C42.223D6, C3⋊C8⋊13D4, C4⋊Q8⋊3S3, C3⋊3(C8⋊5D4), C4.17(S3×D4), C12.37(C2×D4), (C2×Q8).69D6, C4⋊1(Q8⋊2S3), (C2×C12).156D4, C4⋊D12.8C2, C6.77(C2×SD16), C6.23(C4⋊1D4), (C6×Q8).63C22, C2.14(C12⋊3D4), (C2×C12).403C23, (C4×C12).132C22, (C2×D12).107C22, (C4×C3⋊C8)⋊17C2, (C3×C4⋊Q8)⋊3C2, (C2×C6).534(C2×D4), (C2×Q8⋊2S3)⋊15C2, (C2×C3⋊C8).262C22, C2.15(C2×Q8⋊2S3), (C2×C4).136(C3⋊D4), (C2×C4).500(C22×S3), C22.206(C2×C3⋊D4), SmallGroup(192,644)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊6SD16
G = < a,b,c | a12=b8=c2=1, bab-1=a5, cac=a-1, cbc=b3 >
Subgroups: 496 in 142 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C12, D6, C2×C6, C42, C4⋊C4, C2×C8, SD16, C2×D4, C2×Q8, C3⋊C8, D12, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C4×C8, C4⋊1D4, C4⋊Q8, C2×SD16, C2×C3⋊C8, Q8⋊2S3, C4×C12, C3×C4⋊C4, C2×D12, C2×D12, C6×Q8, C8⋊5D4, C4×C3⋊C8, C4⋊D12, C2×Q8⋊2S3, C3×C4⋊Q8, C12⋊6SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C2×SD16, Q8⋊2S3, S3×D4, C2×C3⋊D4, C8⋊5D4, C12⋊3D4, C2×Q8⋊2S3, C12⋊6SD16
►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 14 33 48 94 84 65 53)(2 19 34 41 95 77 66 58)(3 24 35 46 96 82 67 51)(4 17 36 39 85 75 68 56)(5 22 25 44 86 80 69 49)(6 15 26 37 87 73 70 54)(7 20 27 42 88 78 71 59)(8 13 28 47 89 83 72 52)(9 18 29 40 90 76 61 57)(10 23 30 45 91 81 62 50)(11 16 31 38 92 74 63 55)(12 21 32 43 93 79 64 60)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 37)(14 48)(15 47)(16 46)(17 45)(18 44)(19 43)(20 42)(21 41)(22 40)(23 39)(24 38)(25 61)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(49 76)(50 75)(51 74)(52 73)(53 84)(54 83)(55 82)(56 81)(57 80)(58 79)(59 78)(60 77)(85 91)(86 90)(87 89)(92 96)(93 95)
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), (1,14,33,48,94,84,65,53)(2,19,34,41,95,77,66,58)(3,24,35,46,96,82,67,51)(4,17,36,39,85,75,68,56)(5,22,25,44,86,80,69,49)(6,15,26,37,87,73,70,54)(7,20,27,42,88,78,71,59)(8,13,28,47,89,83,72,52)(9,18,29,40,90,76,61,57)(10,23,30,45,91,81,62,50)(11,16,31,38,92,74,63,55)(12,21,32,43,93,79,64,60), (2,12)(3,11)(4,10)(5,9)(6,8)(13,37)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(49,76)(50,75)(51,74)(52,73)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(85,91)(86,90)(87,89)(92,96)(93,95)>;
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), (1,14,33,48,94,84,65,53)(2,19,34,41,95,77,66,58)(3,24,35,46,96,82,67,51)(4,17,36,39,85,75,68,56)(5,22,25,44,86,80,69,49)(6,15,26,37,87,73,70,54)(7,20,27,42,88,78,71,59)(8,13,28,47,89,83,72,52)(9,18,29,40,90,76,61,57)(10,23,30,45,91,81,62,50)(11,16,31,38,92,74,63,55)(12,21,32,43,93,79,64,60), (2,12)(3,11)(4,10)(5,9)(6,8)(13,37)(14,48)(15,47)(16,46)(17,45)(18,44)(19,43)(20,42)(21,41)(22,40)(23,39)(24,38)(25,61)(26,72)(27,71)(28,70)(29,69)(30,68)(31,67)(32,66)(33,65)(34,64)(35,63)(36,62)(49,76)(50,75)(51,74)(52,73)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(85,91)(86,90)(87,89)(92,96)(93,95) );
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)], [(1,14,33,48,94,84,65,53),(2,19,34,41,95,77,66,58),(3,24,35,46,96,82,67,51),(4,17,36,39,85,75,68,56),(5,22,25,44,86,80,69,49),(6,15,26,37,87,73,70,54),(7,20,27,42,88,78,71,59),(8,13,28,47,89,83,72,52),(9,18,29,40,90,76,61,57),(10,23,30,45,91,81,62,50),(11,16,31,38,92,74,63,55),(12,21,32,43,93,79,64,60)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,37),(14,48),(15,47),(16,46),(17,45),(18,44),(19,43),(20,42),(21,41),(22,40),(23,39),(24,38),(25,61),(26,72),(27,71),(28,70),(29,69),(30,68),(31,67),(32,66),(33,65),(34,64),(35,63),(36,62),(49,76),(50,75),(51,74),(52,73),(53,84),(54,83),(55,82),(56,81),(57,80),(58,79),(59,78),(60,77),(85,91),(86,90),(87,89),(92,96),(93,95)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 24 | 24 | 2 | 2 | ··· | 2 | 8 | 8 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | C3⋊D4 | Q8⋊2S3 | S3×D4 |
| kernel | C12⋊6SD16 | C4×C3⋊C8 | C4⋊D12 | C2×Q8⋊2S3 | C3×C4⋊Q8 | C4⋊Q8 | C3⋊C8 | C2×C12 | C42 | C2×Q8 | C12 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 4 | 2 |
Matrix representation of C12⋊6SD16 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 67 | 0 | 0 |
| 0 | 0 | 6 | 67 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 6 |
| 0 | 0 | 0 | 0 | 67 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,67,6,0,0,0,0,67,67,0,0,0,0,0,0,6,67,0,0,0,0,6,6],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;
C12⋊6SD16 in GAP, Magma, Sage, TeX
C_{12}\rtimes_6{\rm SD}_{16} % in TeX
G:=Group("C12:6SD16"); // GroupNames label
G:=SmallGroup(192,644);
// by ID
G=gap.SmallGroup(192,644);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,219,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^3>;
// generators/relations