metabelian, supersoluble, monomial
Aliases: D6:6Dic6, C62.26C23, (S3xC6):6Q8, C6.21(S3xQ8), (C6xDic6):5C2, (C2xDic6):2S3, (C3xC12).73D4, C6.9(C2xDic6), C3:4(D6:3Q8), C6.7(C4oD12), (C2xC12).276D6, C2.11(S3xDic6), C32:4(C22:Q8), C12.51(C3:D4), D6:Dic3.12C2, (C6xC12).90C22, C6.7(Q8:3S3), (C2xDic3).10D6, (C22xS3).62D6, C12:Dic3:16C2, C4.15(D6:S3), C3:4(C12.48D4), C62.C22:21C2, C2.10(D6.6D6), (C6xDic3).75C22, (C2xC4).72S32, (S3xC2xC4).3S3, (S3xC2xC12).6C2, C22.83(C2xS32), (C3xC6).81(C2xD4), C6.75(C2xC3:D4), (C3xC6).17(C2xQ8), (S3xC2xC6).72C22, (C3xC6).14(C4oD4), C2.10(C2xD6:S3), (C2xC6).45(C22xS3), (C2xC3:Dic3).24C22, SmallGroup(288,504)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6:6Dic6
G = < a,b,c,d | a6=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 522 in 161 conjugacy classes, 56 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, Q8, C23, C32, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xQ8, C3xS3, C3xC6, Dic6, C4xS3, C2xDic3, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, C22xC6, C22:Q8, C3xDic3, C3:Dic3, C3xC12, S3xC6, S3xC6, C62, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C2xDic6, S3xC2xC4, C22xC12, C6xQ8, C3xDic6, S3xC12, C6xDic3, C6xDic3, C2xC3:Dic3, C6xC12, S3xC2xC6, C12.48D4, D6:3Q8, D6:Dic3, C62.C22, C12:Dic3, C6xDic6, S3xC2xC12, D6:6Dic6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C4oD4, Dic6, C3:D4, C22xS3, C22:Q8, S32, C2xDic6, C4oD12, S3xQ8, Q8:3S3, C2xC3:D4, D6:S3, C2xS32, C12.48D4, D6:3Q8, S3xDic6, D6.6D6, C2xD6:S3, D6:6Dic6
►On 96 points
(1 87 9 95 5 91)(2 88 10 96 6 92)(3 89 11 85 7 93)(4 90 12 86 8 94)(13 37 17 41 21 45)(14 38 18 42 22 46)(15 39 19 43 23 47)(16 40 20 44 24 48)(25 71 33 67 29 63)(26 72 34 68 30 64)(27 61 35 69 31 65)(28 62 36 70 32 66)(49 78 53 82 57 74)(50 79 54 83 58 75)(51 80 55 84 59 76)(52 81 56 73 60 77)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 47)(26 48)(27 37)(28 38)(29 39)(30 40)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(57 93)(58 94)(59 95)(60 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 38 7 44)(2 37 8 43)(3 48 9 42)(4 47 10 41)(5 46 11 40)(6 45 12 39)(13 86 19 92)(14 85 20 91)(15 96 21 90)(16 95 22 89)(17 94 23 88)(18 93 24 87)(25 56 31 50)(26 55 32 49)(27 54 33 60)(28 53 34 59)(29 52 35 58)(30 51 36 57)(61 83 67 77)(62 82 68 76)(63 81 69 75)(64 80 70 74)(65 79 71 73)(66 78 72 84)
G:=sub<Sym(96)| (1,87,9,95,5,91)(2,88,10,96,6,92)(3,89,11,85,7,93)(4,90,12,86,8,94)(13,37,17,41,21,45)(14,38,18,42,22,46)(15,39,19,43,23,47)(16,40,20,44,24,48)(25,71,33,67,29,63)(26,72,34,68,30,64)(27,61,35,69,31,65)(28,62,36,70,32,66)(49,78,53,82,57,74)(50,79,54,83,58,75)(51,80,55,84,59,76)(52,81,56,73,60,77), (1,80)(2,81)(3,82)(4,83)(5,84)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96), (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), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,86,19,92)(14,85,20,91)(15,96,21,90)(16,95,22,89)(17,94,23,88)(18,93,24,87)(25,56,31,50)(26,55,32,49)(27,54,33,60)(28,53,34,59)(29,52,35,58)(30,51,36,57)(61,83,67,77)(62,82,68,76)(63,81,69,75)(64,80,70,74)(65,79,71,73)(66,78,72,84)>;
G:=Group( (1,87,9,95,5,91)(2,88,10,96,6,92)(3,89,11,85,7,93)(4,90,12,86,8,94)(13,37,17,41,21,45)(14,38,18,42,22,46)(15,39,19,43,23,47)(16,40,20,44,24,48)(25,71,33,67,29,63)(26,72,34,68,30,64)(27,61,35,69,31,65)(28,62,36,70,32,66)(49,78,53,82,57,74)(50,79,54,83,58,75)(51,80,55,84,59,76)(52,81,56,73,60,77), (1,80)(2,81)(3,82)(4,83)(5,84)(6,73)(7,74)(8,75)(9,76)(10,77)(11,78)(12,79)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,47)(26,48)(27,37)(28,38)(29,39)(30,40)(31,41)(32,42)(33,43)(34,44)(35,45)(36,46)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96), (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), (1,38,7,44)(2,37,8,43)(3,48,9,42)(4,47,10,41)(5,46,11,40)(6,45,12,39)(13,86,19,92)(14,85,20,91)(15,96,21,90)(16,95,22,89)(17,94,23,88)(18,93,24,87)(25,56,31,50)(26,55,32,49)(27,54,33,60)(28,53,34,59)(29,52,35,58)(30,51,36,57)(61,83,67,77)(62,82,68,76)(63,81,69,75)(64,80,70,74)(65,79,71,73)(66,78,72,84) );
G=PermutationGroup([[(1,87,9,95,5,91),(2,88,10,96,6,92),(3,89,11,85,7,93),(4,90,12,86,8,94),(13,37,17,41,21,45),(14,38,18,42,22,46),(15,39,19,43,23,47),(16,40,20,44,24,48),(25,71,33,67,29,63),(26,72,34,68,30,64),(27,61,35,69,31,65),(28,62,36,70,32,66),(49,78,53,82,57,74),(50,79,54,83,58,75),(51,80,55,84,59,76),(52,81,56,73,60,77)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,73),(7,74),(8,75),(9,76),(10,77),(11,78),(12,79),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,47),(26,48),(27,37),(28,38),(29,39),(30,40),(31,41),(32,42),(33,43),(34,44),(35,45),(36,46),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96)], [(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)], [(1,38,7,44),(2,37,8,43),(3,48,9,42),(4,47,10,41),(5,46,11,40),(6,45,12,39),(13,86,19,92),(14,85,20,91),(15,96,21,90),(16,95,22,89),(17,94,23,88),(18,93,24,87),(25,56,31,50),(26,55,32,49),(27,54,33,60),(28,53,34,59),(29,52,35,58),(30,51,36,57),(61,83,67,77),(62,82,68,76),(63,81,69,75),(64,80,70,74),(65,79,71,73),(66,78,72,84)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 12O | 12P | 12Q | 12R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 12 | 12 | 36 | 36 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 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 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | Q8 | D6 | D6 | D6 | C4oD4 | C3:D4 | Dic6 | C4oD12 | S32 | S3xQ8 | Q8:3S3 | D6:S3 | C2xS32 | S3xDic6 | D6.6D6 |
| kernel | D6:6Dic6 | D6:Dic3 | C62.C22 | C12:Dic3 | C6xDic6 | S3xC2xC12 | C2xDic6 | S3xC2xC4 | C3xC12 | S3xC6 | C2xDic3 | C2xC12 | C22xS3 | C3xC6 | C12 | D6 | C6 | C2xC4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 1 | 2 | 8 | 4 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of D6:6Dic6 ►in GL6(F13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 4 | 2 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,6,10,0,0,0,0,3,7],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,11,4,0,0,0,0,9,2] >;
D6:6Dic6 in GAP, Magma, Sage, TeX
D_6\rtimes_6{\rm Dic}_6 % in TeX
G:=Group("D6:6Dic6"); // GroupNames label
G:=SmallGroup(288,504);
// by ID
G=gap.SmallGroup(288,504);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,219,100,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations