metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12:7Q8, C42.149D6, C6.1312+ 1+4, C12:Q8:36C2, C4.16(S3xQ8), C4:C4.205D6, C3:7(D4:3Q8), D6.11(C2xQ8), C12.51(C2xQ8), C42.C2:5S3, C4.D12:35C2, D6:Q8:34C2, (C4xDic6):47C2, (C4xD12).24C2, C2.56(D4oD12), C6.43(C22xQ8), (C2xC6).234C24, (C2xC12).88C23, D6:C4.40C22, C4.Dic6:34C2, Dic3:5D4.11C2, (C4xC12).194C22, Dic3.29(C4oD4), (C2xD12).267C22, C4:Dic3.379C22, C22.255(S3xC23), Dic3:C4.144C22, (C22xS3).221C23, (C2xDic3).122C23, (C4xDic3).141C22, (C2xDic6).251C22, (S3xC4:C4):35C2, C2.26(C2xS3xQ8), C2.85(S3xC4oD4), C6.196(C2xC4oD4), (C3xC42.C2):7C2, (S3xC2xC4).217C22, (C2xC4).78(C22xS3), (C3xC4:C4).189C22, SmallGroup(192,1249)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:7Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=a-1, ac=ca, dad-1=a5, cbc-1=a6b, dbd-1=a10b, dcd-1=c-1 >
Subgroups: 560 in 228 conjugacy classes, 105 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2xC4, C2xC4, C2xC4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C42, C42, C22:C4, C4:C4, C4:C4, C4:C4, C22xC4, C2xD4, C2xQ8, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C2xC12, C2xC12, C22xS3, C2xC4:C4, C4xD4, C4xQ8, C22:Q8, C42.C2, C42.C2, C4:Q8, C4xDic3, Dic3:C4, Dic3:C4, C4:Dic3, C4:Dic3, D6:C4, C4xC12, C3xC4:C4, C3xC4:C4, C2xDic6, C2xDic6, S3xC2xC4, C2xD12, D4:3Q8, C4xDic6, C4xD12, C12:Q8, C4.Dic6, S3xC4:C4, Dic3:5D4, D6:Q8, C4.D12, C3xC42.C2, D12:7Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, C4oD4, C24, C22xS3, C22xQ8, C2xC4oD4, 2+ 1+4, S3xQ8, S3xC23, D4:3Q8, C2xS3xQ8, S3xC4oD4, D4oD12, D12:7Q8
►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 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 60)(8 59)(9 58)(10 57)(11 56)(12 55)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 36)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 96)(69 95)(70 94)(71 93)(72 92)
(1 72 58 90)(2 61 59 91)(3 62 60 92)(4 63 49 93)(5 64 50 94)(6 65 51 95)(7 66 52 96)(8 67 53 85)(9 68 54 86)(10 69 55 87)(11 70 56 88)(12 71 57 89)(13 43 25 74)(14 44 26 75)(15 45 27 76)(16 46 28 77)(17 47 29 78)(18 48 30 79)(19 37 31 80)(20 38 32 81)(21 39 33 82)(22 40 34 83)(23 41 35 84)(24 42 36 73)
(1 81 58 38)(2 74 59 43)(3 79 60 48)(4 84 49 41)(5 77 50 46)(6 82 51 39)(7 75 52 44)(8 80 53 37)(9 73 54 42)(10 78 55 47)(11 83 56 40)(12 76 57 45)(13 61 25 91)(14 66 26 96)(15 71 27 89)(16 64 28 94)(17 69 29 87)(18 62 30 92)(19 67 31 85)(20 72 32 90)(21 65 33 95)(22 70 34 88)(23 63 35 93)(24 68 36 86)
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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,36)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,96)(69,95)(70,94)(71,93)(72,92), (1,72,58,90)(2,61,59,91)(3,62,60,92)(4,63,49,93)(5,64,50,94)(6,65,51,95)(7,66,52,96)(8,67,53,85)(9,68,54,86)(10,69,55,87)(11,70,56,88)(12,71,57,89)(13,43,25,74)(14,44,26,75)(15,45,27,76)(16,46,28,77)(17,47,29,78)(18,48,30,79)(19,37,31,80)(20,38,32,81)(21,39,33,82)(22,40,34,83)(23,41,35,84)(24,42,36,73), (1,81,58,38)(2,74,59,43)(3,79,60,48)(4,84,49,41)(5,77,50,46)(6,82,51,39)(7,75,52,44)(8,80,53,37)(9,73,54,42)(10,78,55,47)(11,83,56,40)(12,76,57,45)(13,61,25,91)(14,66,26,96)(15,71,27,89)(16,64,28,94)(17,69,29,87)(18,62,30,92)(19,67,31,85)(20,72,32,90)(21,65,33,95)(22,70,34,88)(23,63,35,93)(24,68,36,86)>;
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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,60)(8,59)(9,58)(10,57)(11,56)(12,55)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,36)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,96)(69,95)(70,94)(71,93)(72,92), (1,72,58,90)(2,61,59,91)(3,62,60,92)(4,63,49,93)(5,64,50,94)(6,65,51,95)(7,66,52,96)(8,67,53,85)(9,68,54,86)(10,69,55,87)(11,70,56,88)(12,71,57,89)(13,43,25,74)(14,44,26,75)(15,45,27,76)(16,46,28,77)(17,47,29,78)(18,48,30,79)(19,37,31,80)(20,38,32,81)(21,39,33,82)(22,40,34,83)(23,41,35,84)(24,42,36,73), (1,81,58,38)(2,74,59,43)(3,79,60,48)(4,84,49,41)(5,77,50,46)(6,82,51,39)(7,75,52,44)(8,80,53,37)(9,73,54,42)(10,78,55,47)(11,83,56,40)(12,76,57,45)(13,61,25,91)(14,66,26,96)(15,71,27,89)(16,64,28,94)(17,69,29,87)(18,62,30,92)(19,67,31,85)(20,72,32,90)(21,65,33,95)(22,70,34,88)(23,63,35,93)(24,68,36,86) );
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)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,60),(8,59),(9,58),(10,57),(11,56),(12,55),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,36),(37,84),(38,83),(39,82),(40,81),(41,80),(42,79),(43,78),(44,77),(45,76),(46,75),(47,74),(48,73),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,96),(69,95),(70,94),(71,93),(72,92)], [(1,72,58,90),(2,61,59,91),(3,62,60,92),(4,63,49,93),(5,64,50,94),(6,65,51,95),(7,66,52,96),(8,67,53,85),(9,68,54,86),(10,69,55,87),(11,70,56,88),(12,71,57,89),(13,43,25,74),(14,44,26,75),(15,45,27,76),(16,46,28,77),(17,47,29,78),(18,48,30,79),(19,37,31,80),(20,38,32,81),(21,39,33,82),(22,40,34,83),(23,41,35,84),(24,42,36,73)], [(1,81,58,38),(2,74,59,43),(3,79,60,48),(4,84,49,41),(5,77,50,46),(6,82,51,39),(7,75,52,44),(8,80,53,37),(9,73,54,42),(10,78,55,47),(11,83,56,40),(12,76,57,45),(13,61,25,91),(14,66,26,96),(15,71,27,89),(16,64,28,94),(17,69,29,87),(18,62,30,92),(19,67,31,85),(20,72,32,90),(21,65,33,95),(22,70,34,88),(23,63,35,93),(24,68,36,86)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | C4oD4 | 2+ 1+4 | S3xQ8 | S3xC4oD4 | D4oD12 |
| kernel | D12:7Q8 | C4xDic6 | C4xD12 | C12:Q8 | C4.Dic6 | S3xC4:C4 | Dic3:5D4 | D6:Q8 | C4.D12 | C3xC42.C2 | C42.C2 | D12 | C42 | C4:C4 | Dic3 | C6 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 1 | 1 | 4 | 1 | 6 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of D12:7Q8 ►in GL6(F13)
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 12 | 12 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 10 | 0 | 0 |
| 0 | 0 | 10 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,4,10,0,0,0,0,10,9,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
D12:7Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_7Q_8 % in TeX
G:=Group("D12:7Q8"); // GroupNames label
G:=SmallGroup(192,1249);
// by ID
G=gap.SmallGroup(192,1249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,219,184,1571,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,c*b*c^-1=a^6*b,d*b*d^-1=a^10*b,d*c*d^-1=c^-1>;
// generators/relations