metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6:10D4, C42.142D6, C6.912- 1+4, C4.71(S3xD4), (C4xD12):44C2, C3:5(Q8:5D4), C12.64(C2xD4), D6:15(C4oD4), Dic3:D4:41C2, D6:3D4:34C2, D6:3Q8:29C2, (C4xDic6):45C2, (C2xD4).174D6, C4.4D4:11S3, (C2xQ8).161D6, C22:C4.73D6, C6.91(C22xD4), (C2xC12).81C23, (C2xC6).221C24, C2.52(Q8oD12), Dic3.28(C2xD4), Dic3:4D4:30C2, (C4xC12).186C22, D6:C4.135C22, (C6xD4).156C22, (C22xC6).51C23, C23.53(C22xS3), (C6xQ8).127C22, Dic3.D4:41C2, C23.11D6:40C2, (C2xD12).223C22, C4:Dic3.377C22, C22.242(S3xC23), Dic3:C4.121C22, (C22xS3).216C23, (C2xDic3).253C23, (C4xDic3).214C22, (C2xDic6).297C22, C6.D4.55C22, (C22xDic3).143C22, (C2xS3xQ8):11C2, C2.64(C2xS3xD4), C2.77(S3xC4oD4), C6.188(C2xC4oD4), (C2xD4:2S3):19C2, (C3xC4.4D4):13C2, (S3xC2xC4).121C22, (C2xC4).196(C22xS3), (C2xC3:D4).60C22, (C3xC22:C4).65C22, SmallGroup(192,1236)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6:10D4
G = < a,b,c,d | a12=c4=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a5, cbc-1=dbd=a6b, dcd=c-1 >
Subgroups: 736 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C42, C42, C22:C4, C22:C4, C4:C4, C22xC4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, Dic6, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xC6, C4xD4, C4xQ8, C4:D4, C22:Q8, C4.4D4, C4.4D4, C22xQ8, C2xC4oD4, C4xDic3, Dic3:C4, C4:Dic3, D6:C4, D6:C4, C6.D4, C4xC12, C3xC22:C4, C2xDic6, C2xDic6, S3xC2xC4, S3xC2xC4, C2xD12, D4:2S3, S3xQ8, C22xDic3, C2xC3:D4, C6xD4, C6xQ8, Q8:5D4, C4xDic6, C4xD12, Dic3.D4, Dic3:4D4, Dic3:D4, C23.11D6, D6:3D4, D6:3Q8, C3xC4.4D4, C2xD4:2S3, C2xS3xQ8, Dic6:10D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, C22xS3, C22xD4, C2xC4oD4, 2- 1+4, S3xD4, S3xC23, Q8:5D4, C2xS3xD4, S3xC4oD4, Q8oD12, Dic6:10D4
►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 65 7 71)(2 64 8 70)(3 63 9 69)(4 62 10 68)(5 61 11 67)(6 72 12 66)(13 28 19 34)(14 27 20 33)(15 26 21 32)(16 25 22 31)(17 36 23 30)(18 35 24 29)(37 86 43 92)(38 85 44 91)(39 96 45 90)(40 95 46 89)(41 94 47 88)(42 93 48 87)(49 83 55 77)(50 82 56 76)(51 81 57 75)(52 80 58 74)(53 79 59 73)(54 78 60 84)
(1 92 77 26)(2 93 78 27)(3 94 79 28)(4 95 80 29)(5 96 81 30)(6 85 82 31)(7 86 83 32)(8 87 84 33)(9 88 73 34)(10 89 74 35)(11 90 75 36)(12 91 76 25)(13 63 41 59)(14 64 42 60)(15 65 43 49)(16 66 44 50)(17 67 45 51)(18 68 46 52)(19 69 47 53)(20 70 48 54)(21 71 37 55)(22 72 38 56)(23 61 39 57)(24 62 40 58)
(1 77)(2 82)(3 75)(4 80)(5 73)(6 78)(7 83)(8 76)(9 81)(10 74)(11 79)(12 84)(13 23)(14 16)(15 21)(17 19)(18 24)(20 22)(25 33)(27 31)(28 36)(30 34)(37 43)(38 48)(39 41)(40 46)(42 44)(45 47)(49 71)(50 64)(51 69)(52 62)(53 67)(54 72)(55 65)(56 70)(57 63)(58 68)(59 61)(60 66)(85 93)(87 91)(88 96)(90 94)
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), (1,65,7,71)(2,64,8,70)(3,63,9,69)(4,62,10,68)(5,61,11,67)(6,72,12,66)(13,28,19,34)(14,27,20,33)(15,26,21,32)(16,25,22,31)(17,36,23,30)(18,35,24,29)(37,86,43,92)(38,85,44,91)(39,96,45,90)(40,95,46,89)(41,94,47,88)(42,93,48,87)(49,83,55,77)(50,82,56,76)(51,81,57,75)(52,80,58,74)(53,79,59,73)(54,78,60,84), (1,92,77,26)(2,93,78,27)(3,94,79,28)(4,95,80,29)(5,96,81,30)(6,85,82,31)(7,86,83,32)(8,87,84,33)(9,88,73,34)(10,89,74,35)(11,90,75,36)(12,91,76,25)(13,63,41,59)(14,64,42,60)(15,65,43,49)(16,66,44,50)(17,67,45,51)(18,68,46,52)(19,69,47,53)(20,70,48,54)(21,71,37,55)(22,72,38,56)(23,61,39,57)(24,62,40,58), (1,77)(2,82)(3,75)(4,80)(5,73)(6,78)(7,83)(8,76)(9,81)(10,74)(11,79)(12,84)(13,23)(14,16)(15,21)(17,19)(18,24)(20,22)(25,33)(27,31)(28,36)(30,34)(37,43)(38,48)(39,41)(40,46)(42,44)(45,47)(49,71)(50,64)(51,69)(52,62)(53,67)(54,72)(55,65)(56,70)(57,63)(58,68)(59,61)(60,66)(85,93)(87,91)(88,96)(90,94)>;
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), (1,65,7,71)(2,64,8,70)(3,63,9,69)(4,62,10,68)(5,61,11,67)(6,72,12,66)(13,28,19,34)(14,27,20,33)(15,26,21,32)(16,25,22,31)(17,36,23,30)(18,35,24,29)(37,86,43,92)(38,85,44,91)(39,96,45,90)(40,95,46,89)(41,94,47,88)(42,93,48,87)(49,83,55,77)(50,82,56,76)(51,81,57,75)(52,80,58,74)(53,79,59,73)(54,78,60,84), (1,92,77,26)(2,93,78,27)(3,94,79,28)(4,95,80,29)(5,96,81,30)(6,85,82,31)(7,86,83,32)(8,87,84,33)(9,88,73,34)(10,89,74,35)(11,90,75,36)(12,91,76,25)(13,63,41,59)(14,64,42,60)(15,65,43,49)(16,66,44,50)(17,67,45,51)(18,68,46,52)(19,69,47,53)(20,70,48,54)(21,71,37,55)(22,72,38,56)(23,61,39,57)(24,62,40,58), (1,77)(2,82)(3,75)(4,80)(5,73)(6,78)(7,83)(8,76)(9,81)(10,74)(11,79)(12,84)(13,23)(14,16)(15,21)(17,19)(18,24)(20,22)(25,33)(27,31)(28,36)(30,34)(37,43)(38,48)(39,41)(40,46)(42,44)(45,47)(49,71)(50,64)(51,69)(52,62)(53,67)(54,72)(55,65)(56,70)(57,63)(58,68)(59,61)(60,66)(85,93)(87,91)(88,96)(90,94) );
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)], [(1,65,7,71),(2,64,8,70),(3,63,9,69),(4,62,10,68),(5,61,11,67),(6,72,12,66),(13,28,19,34),(14,27,20,33),(15,26,21,32),(16,25,22,31),(17,36,23,30),(18,35,24,29),(37,86,43,92),(38,85,44,91),(39,96,45,90),(40,95,46,89),(41,94,47,88),(42,93,48,87),(49,83,55,77),(50,82,56,76),(51,81,57,75),(52,80,58,74),(53,79,59,73),(54,78,60,84)], [(1,92,77,26),(2,93,78,27),(3,94,79,28),(4,95,80,29),(5,96,81,30),(6,85,82,31),(7,86,83,32),(8,87,84,33),(9,88,73,34),(10,89,74,35),(11,90,75,36),(12,91,76,25),(13,63,41,59),(14,64,42,60),(15,65,43,49),(16,66,44,50),(17,67,45,51),(18,68,46,52),(19,69,47,53),(20,70,48,54),(21,71,37,55),(22,72,38,56),(23,61,39,57),(24,62,40,58)], [(1,77),(2,82),(3,75),(4,80),(5,73),(6,78),(7,83),(8,76),(9,81),(10,74),(11,79),(12,84),(13,23),(14,16),(15,21),(17,19),(18,24),(20,22),(25,33),(27,31),(28,36),(30,34),(37,43),(38,48),(39,41),(40,46),(42,44),(45,47),(49,71),(50,64),(51,69),(52,62),(53,67),(54,72),(55,65),(56,70),(57,63),(58,68),(59,61),(60,66),(85,93),(87,91),(88,96),(90,94)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 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 | D6 | D6 | D6 | D6 | C4oD4 | 2- 1+4 | S3xD4 | S3xC4oD4 | Q8oD12 |
| kernel | Dic6:10D4 | C4xDic6 | C4xD12 | Dic3.D4 | Dic3:4D4 | Dic3:D4 | C23.11D6 | D6:3D4 | D6:3Q8 | C3xC4.4D4 | C2xD4:2S3 | C2xS3xQ8 | C4.4D4 | Dic6 | C42 | C22:C4 | C2xD4 | C2xQ8 | D6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 4 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of Dic6:10D4 ►in GL6(F13)
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 | 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 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 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 | 1 | 2 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(13))| [8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,2,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1] >;
Dic6:10D4 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_{10}D_4 % in TeX
G:=Group("Dic6:10D4"); // GroupNames label
G:=SmallGroup(192,1236);
// by ID
G=gap.SmallGroup(192,1236);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,1571,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=d^2=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^5,c*b*c^-1=d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations