metabelian, supersoluble, monomial
Aliases: C12⋊3Dic6, Dic3⋊4Dic6, Dic3.5D12, C62.88C23, (C3×C12)⋊3Q8, C3⋊1(C12⋊Q8), C6.25(S3×D4), C32⋊7(C4⋊Q8), C6.31(S3×Q8), (C3×Dic3)⋊4Q8, C6.26(C2×D12), C2.27(S3×D12), C4⋊Dic3.7S3, C3⋊1(C12⋊2Q8), (C2×C12).283D6, C4⋊1(C32⋊2Q8), (C4×Dic3).5S3, C6.19(C2×Dic6), C2.19(S3×Dic6), (C3×Dic3).24D4, (Dic3×C12).9C2, (C2×Dic3).36D6, Dic3⋊Dic3.3C2, (C6×C12).110C22, C12⋊Dic3.17C2, (C6×Dic3).20C22, (C2×C4).84S32, (C3×C6).59(C2×D4), (C3×C6).39(C2×Q8), C22.125(C2×S32), C2.6(C2×C32⋊2Q8), (C3×C4⋊Dic3).16C2, (C2×C32⋊2Q8).3C2, (C2×C6).107(C22×S3), (C2×C3⋊Dic3).57C22, SmallGroup(288,566)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊3Dic6
G = < a,b,c | a12=b12=1, c2=b6, bab-1=a-1, ac=ca, cbc-1=b-1 >
Subgroups: 522 in 149 conjugacy classes, 60 normal (34 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, C4⋊C4, C2×Q8, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C4⋊Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C2×Dic6, C32⋊2Q8, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C12⋊2Q8, C12⋊Q8, Dic3⋊Dic3, Dic3×C12, C3×C4⋊Dic3, C12⋊Dic3, C2×C32⋊2Q8, C12⋊3Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, Dic6, D12, C22×S3, C4⋊Q8, S32, C2×Dic6, C2×D12, S3×D4, S3×Q8, C32⋊2Q8, C2×S32, C12⋊2Q8, C12⋊Q8, S3×Dic6, S3×D12, C2×C32⋊2Q8, C12⋊3Dic6
►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 17 26 95 9 21 34 87 5 13 30 91)(2 16 27 94 10 20 35 86 6 24 31 90)(3 15 28 93 11 19 36 85 7 23 32 89)(4 14 29 92 12 18 25 96 8 22 33 88)(37 68 49 73 45 72 57 77 41 64 53 81)(38 67 50 84 46 71 58 76 42 63 54 80)(39 66 51 83 47 70 59 75 43 62 55 79)(40 65 52 82 48 69 60 74 44 61 56 78)
(1 74 34 65)(2 75 35 66)(3 76 36 67)(4 77 25 68)(5 78 26 69)(6 79 27 70)(7 80 28 71)(8 81 29 72)(9 82 30 61)(10 83 31 62)(11 84 32 63)(12 73 33 64)(13 56 95 48)(14 57 96 37)(15 58 85 38)(16 59 86 39)(17 60 87 40)(18 49 88 41)(19 50 89 42)(20 51 90 43)(21 52 91 44)(22 53 92 45)(23 54 93 46)(24 55 94 47)
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), (1,17,26,95,9,21,34,87,5,13,30,91)(2,16,27,94,10,20,35,86,6,24,31,90)(3,15,28,93,11,19,36,85,7,23,32,89)(4,14,29,92,12,18,25,96,8,22,33,88)(37,68,49,73,45,72,57,77,41,64,53,81)(38,67,50,84,46,71,58,76,42,63,54,80)(39,66,51,83,47,70,59,75,43,62,55,79)(40,65,52,82,48,69,60,74,44,61,56,78), (1,74,34,65)(2,75,35,66)(3,76,36,67)(4,77,25,68)(5,78,26,69)(6,79,27,70)(7,80,28,71)(8,81,29,72)(9,82,30,61)(10,83,31,62)(11,84,32,63)(12,73,33,64)(13,56,95,48)(14,57,96,37)(15,58,85,38)(16,59,86,39)(17,60,87,40)(18,49,88,41)(19,50,89,42)(20,51,90,43)(21,52,91,44)(22,53,92,45)(23,54,93,46)(24,55,94,47)>;
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), (1,17,26,95,9,21,34,87,5,13,30,91)(2,16,27,94,10,20,35,86,6,24,31,90)(3,15,28,93,11,19,36,85,7,23,32,89)(4,14,29,92,12,18,25,96,8,22,33,88)(37,68,49,73,45,72,57,77,41,64,53,81)(38,67,50,84,46,71,58,76,42,63,54,80)(39,66,51,83,47,70,59,75,43,62,55,79)(40,65,52,82,48,69,60,74,44,61,56,78), (1,74,34,65)(2,75,35,66)(3,76,36,67)(4,77,25,68)(5,78,26,69)(6,79,27,70)(7,80,28,71)(8,81,29,72)(9,82,30,61)(10,83,31,62)(11,84,32,63)(12,73,33,64)(13,56,95,48)(14,57,96,37)(15,58,85,38)(16,59,86,39)(17,60,87,40)(18,49,88,41)(19,50,89,42)(20,51,90,43)(21,52,91,44)(22,53,92,45)(23,54,93,46)(24,55,94,47) );
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)], [(1,17,26,95,9,21,34,87,5,13,30,91),(2,16,27,94,10,20,35,86,6,24,31,90),(3,15,28,93,11,19,36,85,7,23,32,89),(4,14,29,92,12,18,25,96,8,22,33,88),(37,68,49,73,45,72,57,77,41,64,53,81),(38,67,50,84,46,71,58,76,42,63,54,80),(39,66,51,83,47,70,59,75,43,62,55,79),(40,65,52,82,48,69,60,74,44,61,56,78)], [(1,74,34,65),(2,75,35,66),(3,76,36,67),(4,77,25,68),(5,78,26,69),(6,79,27,70),(7,80,28,71),(8,81,29,72),(9,82,30,61),(10,83,31,62),(11,84,32,63),(12,73,33,64),(13,56,95,48),(14,57,96,37),(15,58,85,38),(16,59,86,39),(17,60,87,40),(18,49,88,41),(19,50,89,42),(20,51,90,43),(21,52,91,44),(22,53,92,45),(23,54,93,46),(24,55,94,47)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | 6H | 6I | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | ··· | 12R | 12S | 12T | 12U | 12V |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | + | - | + | - | + | + | - | - | + | - | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | Q8 | Q8 | D6 | D6 | Dic6 | D12 | Dic6 | S32 | S3×D4 | S3×Q8 | C32⋊2Q8 | C2×S32 | S3×Dic6 | S3×D12 |
| kernel | C12⋊3Dic6 | Dic3⋊Dic3 | Dic3×C12 | C3×C4⋊Dic3 | C12⋊Dic3 | C2×C32⋊2Q8 | C4×Dic3 | C4⋊Dic3 | C3×Dic3 | C3×Dic3 | C3×C12 | C2×Dic3 | C2×C12 | Dic3 | Dic3 | C12 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C12⋊3Dic6 ►in GL6(𝔽13)
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 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 | 1 | 0 |
G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,9,0,0,0,0,0,0,6,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C12⋊3Dic6 in GAP, Magma, Sage, TeX
C_{12}\rtimes_3{\rm Dic}_6 % in TeX
G:=Group("C12:3Dic6"); // GroupNames label
G:=SmallGroup(288,566);
// by ID
G=gap.SmallGroup(288,566);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,64,219,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^12=b^12=1,c^2=b^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations