metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6:21D4, C6.762- 1+4, C3:3(D4xQ8), D6:6(C2xQ8), C3:D4:1Q8, C12:Q8:25C2, C22:Q8:9S3, C22:2(S3xQ8), C4:C4.190D6, Dic3:4(C2xQ8), C4.113(S3xD4), D6:Q8:19C2, C4.D12:26C2, C12.236(C2xD4), (C2xQ8).151D6, C22:C4.58D6, C6.78(C22xD4), C6.35(C22xQ8), (C2xC6).176C24, (C2xC12).55C23, C2.36(Q8oD12), Dic3.24(C2xD4), (C22xC4).254D6, Dic6:C4:25C2, Dic3:Q8:15C2, D6:C4.107C22, Dic3:4D4.1C2, (C22xDic6):17C2, (C6xQ8).108C22, Dic3.D4:23C2, Dic3:C4.28C22, C4:Dic3.216C22, (C22xC6).204C23, C22.197(S3xC23), C23.200(C22xS3), (C22xS3).198C23, (C22xC12).256C22, (C2xDic3).235C23, (C2xDic6).248C22, (C4xDic3).106C22, C6.D4.117C22, (C22xDic3).118C22, (C2xS3xQ8):7C2, (C2xC6):3(C2xQ8), C2.51(C2xS3xD4), C2.18(C2xS3xQ8), (C4xC3:D4).7C2, (S3xC2xC4).96C22, (C3xC22:Q8):12C2, (C2xC4).49(C22xS3), (C3xC4:C4).159C22, (C2xC3:D4).124C22, (C3xC22:C4).31C22, SmallGroup(192,1191)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6:21D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, cac-1=a5, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 656 in 280 conjugacy classes, 115 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xC6, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C22xC4, C2xD4, C2xQ8, C2xQ8, Dic6, Dic6, C4xS3, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C2xC12, C3xQ8, C22xS3, C22xC6, C4xD4, C4xQ8, C22:Q8, C22:Q8, C4:Q8, C22xQ8, C4xDic3, C4xDic3, Dic3:C4, Dic3:C4, C4:Dic3, D6:C4, D6:C4, C6.D4, C3xC22:C4, C3xC4:C4, C3xC4:C4, C2xDic6, C2xDic6, C2xDic6, S3xC2xC4, S3xC2xC4, S3xQ8, C22xDic3, C2xC3:D4, C22xC12, C6xQ8, D4xQ8, Dic3.D4, Dic3:4D4, Dic6:C4, C12:Q8, D6:Q8, C4.D12, C4xC3:D4, Dic3:Q8, C3xC22:Q8, C22xDic6, C2xS3xQ8, Dic6:21D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C24, C22xS3, C22xD4, C22xQ8, 2- 1+4, S3xD4, S3xQ8, S3xC23, D4xQ8, C2xS3xD4, C2xS3xQ8, Q8oD12, Dic6:21D4
►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 56 7 50)(2 55 8 49)(3 54 9 60)(4 53 10 59)(5 52 11 58)(6 51 12 57)(13 74 19 80)(14 73 20 79)(15 84 21 78)(16 83 22 77)(17 82 23 76)(18 81 24 75)(25 41 31 47)(26 40 32 46)(27 39 33 45)(28 38 34 44)(29 37 35 43)(30 48 36 42)(61 87 67 93)(62 86 68 92)(63 85 69 91)(64 96 70 90)(65 95 71 89)(66 94 72 88)
(1 21 95 41)(2 14 96 46)(3 19 85 39)(4 24 86 44)(5 17 87 37)(6 22 88 42)(7 15 89 47)(8 20 90 40)(9 13 91 45)(10 18 92 38)(11 23 93 43)(12 16 94 48)(25 50 84 65)(26 55 73 70)(27 60 74 63)(28 53 75 68)(29 58 76 61)(30 51 77 66)(31 56 78 71)(32 49 79 64)(33 54 80 69)(34 59 81 62)(35 52 82 67)(36 57 83 72)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 37)(10 38)(11 39)(12 40)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 85)(24 86)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 49)(61 80)(62 81)(63 82)(64 83)(65 84)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
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), (1,56,7,50)(2,55,8,49)(3,54,9,60)(4,53,10,59)(5,52,11,58)(6,51,12,57)(13,74,19,80)(14,73,20,79)(15,84,21,78)(16,83,22,77)(17,82,23,76)(18,81,24,75)(25,41,31,47)(26,40,32,46)(27,39,33,45)(28,38,34,44)(29,37,35,43)(30,48,36,42)(61,87,67,93)(62,86,68,92)(63,85,69,91)(64,96,70,90)(65,95,71,89)(66,94,72,88), (1,21,95,41)(2,14,96,46)(3,19,85,39)(4,24,86,44)(5,17,87,37)(6,22,88,42)(7,15,89,47)(8,20,90,40)(9,13,91,45)(10,18,92,38)(11,23,93,43)(12,16,94,48)(25,50,84,65)(26,55,73,70)(27,60,74,63)(28,53,75,68)(29,58,76,61)(30,51,77,66)(31,56,78,71)(32,49,79,64)(33,54,80,69)(34,59,81,62)(35,52,82,67)(36,57,83,72), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,85)(24,86)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,49)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79)>;
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), (1,56,7,50)(2,55,8,49)(3,54,9,60)(4,53,10,59)(5,52,11,58)(6,51,12,57)(13,74,19,80)(14,73,20,79)(15,84,21,78)(16,83,22,77)(17,82,23,76)(18,81,24,75)(25,41,31,47)(26,40,32,46)(27,39,33,45)(28,38,34,44)(29,37,35,43)(30,48,36,42)(61,87,67,93)(62,86,68,92)(63,85,69,91)(64,96,70,90)(65,95,71,89)(66,94,72,88), (1,21,95,41)(2,14,96,46)(3,19,85,39)(4,24,86,44)(5,17,87,37)(6,22,88,42)(7,15,89,47)(8,20,90,40)(9,13,91,45)(10,18,92,38)(11,23,93,43)(12,16,94,48)(25,50,84,65)(26,55,73,70)(27,60,74,63)(28,53,75,68)(29,58,76,61)(30,51,77,66)(31,56,78,71)(32,49,79,64)(33,54,80,69)(34,59,81,62)(35,52,82,67)(36,57,83,72), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,37)(10,38)(11,39)(12,40)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,85)(24,86)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,49)(61,80)(62,81)(63,82)(64,83)(65,84)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79) );
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)], [(1,56,7,50),(2,55,8,49),(3,54,9,60),(4,53,10,59),(5,52,11,58),(6,51,12,57),(13,74,19,80),(14,73,20,79),(15,84,21,78),(16,83,22,77),(17,82,23,76),(18,81,24,75),(25,41,31,47),(26,40,32,46),(27,39,33,45),(28,38,34,44),(29,37,35,43),(30,48,36,42),(61,87,67,93),(62,86,68,92),(63,85,69,91),(64,96,70,90),(65,95,71,89),(66,94,72,88)], [(1,21,95,41),(2,14,96,46),(3,19,85,39),(4,24,86,44),(5,17,87,37),(6,22,88,42),(7,15,89,47),(8,20,90,40),(9,13,91,45),(10,18,92,38),(11,23,93,43),(12,16,94,48),(25,50,84,65),(26,55,73,70),(27,60,74,63),(28,53,75,68),(29,58,76,61),(30,51,77,66),(31,56,78,71),(32,49,79,64),(33,54,80,69),(34,59,81,62),(35,52,82,67),(36,57,83,72)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,37),(10,38),(11,39),(12,40),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,85),(24,86),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,49),(61,80),(62,81),(63,82),(64,83),(65,84),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | D6 | 2- 1+4 | S3xD4 | S3xQ8 | Q8oD12 |
| kernel | Dic6:21D4 | Dic3.D4 | Dic3:4D4 | Dic6:C4 | C12:Q8 | D6:Q8 | C4.D12 | C4xC3:D4 | Dic3:Q8 | C3xC22:Q8 | C22xDic6 | C2xS3xQ8 | C22:Q8 | Dic6 | C3:D4 | C22:C4 | C4:C4 | C22xC4 | C2xQ8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 3 | 1 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of Dic6:21D4 ►in GL6(F13)
| 3 | 9 | 0 | 0 | 0 | 0 |
| 9 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 11 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(13))| [3,9,0,0,0,0,9,10,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,11,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,11,1] >;
Dic6:21D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_{21}D_4 % in TeX
G:=Group("Dic6:21D4"); // GroupNames label
G:=SmallGroup(192,1191);
// by ID
G=gap.SmallGroup(192,1191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,570,185,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^5,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations