metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8:6D12, C42.128D6, C6.112- 1+4, (C4xQ8):13S3, (C3xQ8):10D4, (C4xD12):38C2, C4:C4.295D6, (Q8xC12):11C2, C3:2(Q8:5D4), C4.25(C2xD12), C12.57(C2xD4), D6:13(C4oD4), C12:D4:17C2, C4.D12:18C2, D6:C4.6C22, (C2xQ8).228D6, C6.19(C22xD4), C42:7S3:20C2, (C2xC6).120C24, C2.21(C22xD12), (C4xC12).172C22, (C2xC12).498C23, (C6xQ8).220C22, (C2xD12).216C22, (C22xS3).45C23, C4:Dic3.306C22, C22.141(S3xC23), (C2xDic3).54C23, C2.12(Q8.15D6), (C2xDic6).149C22, (C2xS3xQ8):4C2, C2.29(S3xC4oD4), (C2xQ8:3S3):3C2, C6.145(C2xC4oD4), (S3xC2xC4).72C22, (C3xC4:C4).348C22, (C2xC4).168(C22xS3), SmallGroup(192,1135)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8:6D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 776 in 290 conjugacy classes, 113 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2xC4, C2xC4, C2xC4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C42, C22:C4, C4:C4, C4:C4, C22xC4, C2xD4, C2xQ8, C2xQ8, C4oD4, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, C22xS3, C4xD4, C4xQ8, C4:D4, C22:Q8, C4.4D4, C22xQ8, C2xC4oD4, C4:Dic3, D6:C4, D6:C4, C4xC12, C3xC4:C4, C2xDic6, S3xC2xC4, C2xD12, S3xQ8, Q8:3S3, C6xQ8, Q8:5D4, C4xD12, C42:7S3, C12:D4, C4.D12, Q8xC12, C2xS3xQ8, C2xQ8:3S3, Q8:6D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C24, D12, C22xS3, C22xD4, C2xC4oD4, 2- 1+4, C2xD12, S3xC23, Q8:5D4, C22xD12, Q8.15D6, S3xC4oD4, Q8:6D12
►On 96 points
(1 28 71 14)(2 29 72 15)(3 30 61 16)(4 31 62 17)(5 32 63 18)(6 33 64 19)(7 34 65 20)(8 35 66 21)(9 36 67 22)(10 25 68 23)(11 26 69 24)(12 27 70 13)(37 50 80 86)(38 51 81 87)(39 52 82 88)(40 53 83 89)(41 54 84 90)(42 55 73 91)(43 56 74 92)(44 57 75 93)(45 58 76 94)(46 59 77 95)(47 60 78 96)(48 49 79 85)
(1 82 71 39)(2 40 72 83)(3 84 61 41)(4 42 62 73)(5 74 63 43)(6 44 64 75)(7 76 65 45)(8 46 66 77)(9 78 67 47)(10 48 68 79)(11 80 69 37)(12 38 70 81)(13 51 27 87)(14 88 28 52)(15 53 29 89)(16 90 30 54)(17 55 31 91)(18 92 32 56)(19 57 33 93)(20 94 34 58)(21 59 35 95)(22 96 36 60)(23 49 25 85)(24 86 26 50)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 23)(14 22)(15 21)(16 20)(17 19)(25 27)(28 36)(29 35)(30 34)(31 33)(37 80)(38 79)(39 78)(40 77)(41 76)(42 75)(43 74)(44 73)(45 84)(46 83)(47 82)(48 81)(49 87)(50 86)(51 85)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(61 65)(62 64)(66 72)(67 71)(68 70)
G:=sub<Sym(96)| (1,28,71,14)(2,29,72,15)(3,30,61,16)(4,31,62,17)(5,32,63,18)(6,33,64,19)(7,34,65,20)(8,35,66,21)(9,36,67,22)(10,25,68,23)(11,26,69,24)(12,27,70,13)(37,50,80,86)(38,51,81,87)(39,52,82,88)(40,53,83,89)(41,54,84,90)(42,55,73,91)(43,56,74,92)(44,57,75,93)(45,58,76,94)(46,59,77,95)(47,60,78,96)(48,49,79,85), (1,82,71,39)(2,40,72,83)(3,84,61,41)(4,42,62,73)(5,74,63,43)(6,44,64,75)(7,76,65,45)(8,46,66,77)(9,78,67,47)(10,48,68,79)(11,80,69,37)(12,38,70,81)(13,51,27,87)(14,88,28,52)(15,53,29,89)(16,90,30,54)(17,55,31,91)(18,92,32,56)(19,57,33,93)(20,94,34,58)(21,59,35,95)(22,96,36,60)(23,49,25,85)(24,86,26,50), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,23)(14,22)(15,21)(16,20)(17,19)(25,27)(28,36)(29,35)(30,34)(31,33)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,84)(46,83)(47,82)(48,81)(49,87)(50,86)(51,85)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,65)(62,64)(66,72)(67,71)(68,70)>;
G:=Group( (1,28,71,14)(2,29,72,15)(3,30,61,16)(4,31,62,17)(5,32,63,18)(6,33,64,19)(7,34,65,20)(8,35,66,21)(9,36,67,22)(10,25,68,23)(11,26,69,24)(12,27,70,13)(37,50,80,86)(38,51,81,87)(39,52,82,88)(40,53,83,89)(41,54,84,90)(42,55,73,91)(43,56,74,92)(44,57,75,93)(45,58,76,94)(46,59,77,95)(47,60,78,96)(48,49,79,85), (1,82,71,39)(2,40,72,83)(3,84,61,41)(4,42,62,73)(5,74,63,43)(6,44,64,75)(7,76,65,45)(8,46,66,77)(9,78,67,47)(10,48,68,79)(11,80,69,37)(12,38,70,81)(13,51,27,87)(14,88,28,52)(15,53,29,89)(16,90,30,54)(17,55,31,91)(18,92,32,56)(19,57,33,93)(20,94,34,58)(21,59,35,95)(22,96,36,60)(23,49,25,85)(24,86,26,50), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,23)(14,22)(15,21)(16,20)(17,19)(25,27)(28,36)(29,35)(30,34)(31,33)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,84)(46,83)(47,82)(48,81)(49,87)(50,86)(51,85)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,65)(62,64)(66,72)(67,71)(68,70) );
G=PermutationGroup([[(1,28,71,14),(2,29,72,15),(3,30,61,16),(4,31,62,17),(5,32,63,18),(6,33,64,19),(7,34,65,20),(8,35,66,21),(9,36,67,22),(10,25,68,23),(11,26,69,24),(12,27,70,13),(37,50,80,86),(38,51,81,87),(39,52,82,88),(40,53,83,89),(41,54,84,90),(42,55,73,91),(43,56,74,92),(44,57,75,93),(45,58,76,94),(46,59,77,95),(47,60,78,96),(48,49,79,85)], [(1,82,71,39),(2,40,72,83),(3,84,61,41),(4,42,62,73),(5,74,63,43),(6,44,64,75),(7,76,65,45),(8,46,66,77),(9,78,67,47),(10,48,68,79),(11,80,69,37),(12,38,70,81),(13,51,27,87),(14,88,28,52),(15,53,29,89),(16,90,30,54),(17,55,31,91),(18,92,32,56),(19,57,33,93),(20,94,34,58),(21,59,35,95),(22,96,36,60),(23,49,25,85),(24,86,26,50)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,23),(14,22),(15,21),(16,20),(17,19),(25,27),(28,36),(29,35),(30,34),(31,33),(37,80),(38,79),(39,78),(40,77),(41,76),(42,75),(43,74),(44,73),(45,84),(46,83),(47,82),(48,81),(49,87),(50,86),(51,85),(52,96),(53,95),(54,94),(55,93),(56,92),(57,91),(58,90),(59,89),(60,88),(61,65),(62,64),(66,72),(67,71),(68,70)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4oD4 | D12 | 2- 1+4 | Q8.15D6 | S3xC4oD4 |
| kernel | Q8:6D12 | C4xD12 | C42:7S3 | C12:D4 | C4.D12 | Q8xC12 | C2xS3xQ8 | C2xQ8:3S3 | C4xQ8 | C3xQ8 | C42 | C4:C4 | C2xQ8 | D6 | Q8 | C6 | C2 | C2 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of Q8:6D12 ►in GL4(F13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
| 6 | 10 | 0 | 0 |
| 3 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,5,0,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,0,12,0,0,1,0],[6,3,0,0,10,3,0,0,0,0,1,0,0,0,0,12],[12,0,0,0,1,1,0,0,0,0,1,0,0,0,0,12] >;
Q8:6D12 in GAP, Magma, Sage, TeX
Q_8\rtimes_6D_{12} % in TeX
G:=Group("Q8:6D12"); // GroupNames label
G:=SmallGroup(192,1135);
// by ID
G=gap.SmallGroup(192,1135);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations