metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8:3D12, D6:7SD16, Dic6:4D4, (C3xQ8):1D4, D6:C8:13C2, C4:C4.23D6, C4.5(C2xD12), C4.92(S3xD4), C3:2(Q8:D4), (C2xC8).123D6, Q8:C4:11S3, C6.23C22wrC2, C12:D4.1C2, C12.120(C2xD4), (C2xQ8).131D6, C6.31(C2xSD16), C2.17(S3xSD16), C6.SD16:11C2, (C2xDic3).30D4, (C22xS3).77D4, C22.196(S3xD4), (C6xQ8).29C22, C2.26(D6:D4), (C2xC24).134C22, (C2xC12).246C23, C2.14(Q16:S3), (C2xD12).62C22, C6.60(C8.C22), (C2xDic6).70C22, (C2xS3xQ8):1C2, (C2xC24:C2):17C2, (C2xQ8:2S3):2C2, (C2xC6).259(C2xD4), (C2xC3:C8).38C22, (S3xC2xC4).20C22, (C3xQ8:C4):11C2, (C3xC4:C4).47C22, (C2xC4).353(C22xS3), SmallGroup(192,365)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8:3D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=dbd=ab, dcd=c-1 >
Subgroups: 520 in 158 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2xC6, C22:C4, C4:C4, C2xC8, C2xC8, SD16, C22xC4, C2xD4, C2xQ8, C2xQ8, C3:C8, C24, Dic6, Dic6, C4xS3, D12, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xQ8, C3xQ8, C22xS3, C22xS3, C22:C8, Q8:C4, Q8:C4, C4:D4, C2xSD16, C22xQ8, C24:C2, C2xC3:C8, D6:C4, Q8:2S3, C3xC4:C4, C2xC24, C2xDic6, C2xDic6, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, S3xQ8, C6xQ8, Q8:D4, C6.SD16, D6:C8, C3xQ8:C4, C12:D4, C2xC24:C2, C2xQ8:2S3, C2xS3xQ8, Q8:3D12
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, D12, C22xS3, C22wrC2, C2xSD16, C8.C22, C2xD12, S3xD4, Q8:D4, D6:D4, S3xSD16, Q16:S3, Q8:3D12
►On 96 points
(1 57 37 23)(2 24 38 58)(3 59 39 13)(4 14 40 60)(5 49 41 15)(6 16 42 50)(7 51 43 17)(8 18 44 52)(9 53 45 19)(10 20 46 54)(11 55 47 21)(12 22 48 56)(25 86 83 72)(26 61 84 87)(27 88 73 62)(28 63 74 89)(29 90 75 64)(30 65 76 91)(31 92 77 66)(32 67 78 93)(33 94 79 68)(34 69 80 95)(35 96 81 70)(36 71 82 85)
(1 86 37 72)(2 26 38 84)(3 88 39 62)(4 28 40 74)(5 90 41 64)(6 30 42 76)(7 92 43 66)(8 32 44 78)(9 94 45 68)(10 34 46 80)(11 96 47 70)(12 36 48 82)(13 73 59 27)(14 89 60 63)(15 75 49 29)(16 91 50 65)(17 77 51 31)(18 93 52 67)(19 79 53 33)(20 95 54 69)(21 81 55 35)(22 85 56 71)(23 83 57 25)(24 87 58 61)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 60)(23 59)(24 58)(25 88)(26 87)(27 86)(28 85)(29 96)(30 95)(31 94)(32 93)(33 92)(34 91)(35 90)(36 89)(37 39)(40 48)(41 47)(42 46)(43 45)(61 84)(62 83)(63 82)(64 81)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)(71 74)(72 73)
G:=sub<Sym(96)| (1,57,37,23)(2,24,38,58)(3,59,39,13)(4,14,40,60)(5,49,41,15)(6,16,42,50)(7,51,43,17)(8,18,44,52)(9,53,45,19)(10,20,46,54)(11,55,47,21)(12,22,48,56)(25,86,83,72)(26,61,84,87)(27,88,73,62)(28,63,74,89)(29,90,75,64)(30,65,76,91)(31,92,77,66)(32,67,78,93)(33,94,79,68)(34,69,80,95)(35,96,81,70)(36,71,82,85), (1,86,37,72)(2,26,38,84)(3,88,39,62)(4,28,40,74)(5,90,41,64)(6,30,42,76)(7,92,43,66)(8,32,44,78)(9,94,45,68)(10,34,46,80)(11,96,47,70)(12,36,48,82)(13,73,59,27)(14,89,60,63)(15,75,49,29)(16,91,50,65)(17,77,51,31)(18,93,52,67)(19,79,53,33)(20,95,54,69)(21,81,55,35)(22,85,56,71)(23,83,57,25)(24,87,58,61), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,39)(40,48)(41,47)(42,46)(43,45)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)>;
G:=Group( (1,57,37,23)(2,24,38,58)(3,59,39,13)(4,14,40,60)(5,49,41,15)(6,16,42,50)(7,51,43,17)(8,18,44,52)(9,53,45,19)(10,20,46,54)(11,55,47,21)(12,22,48,56)(25,86,83,72)(26,61,84,87)(27,88,73,62)(28,63,74,89)(29,90,75,64)(30,65,76,91)(31,92,77,66)(32,67,78,93)(33,94,79,68)(34,69,80,95)(35,96,81,70)(36,71,82,85), (1,86,37,72)(2,26,38,84)(3,88,39,62)(4,28,40,74)(5,90,41,64)(6,30,42,76)(7,92,43,66)(8,32,44,78)(9,94,45,68)(10,34,46,80)(11,96,47,70)(12,36,48,82)(13,73,59,27)(14,89,60,63)(15,75,49,29)(16,91,50,65)(17,77,51,31)(18,93,52,67)(19,79,53,33)(20,95,54,69)(21,81,55,35)(22,85,56,71)(23,83,57,25)(24,87,58,61), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,88)(26,87)(27,86)(28,85)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,39)(40,48)(41,47)(42,46)(43,45)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73) );
G=PermutationGroup([[(1,57,37,23),(2,24,38,58),(3,59,39,13),(4,14,40,60),(5,49,41,15),(6,16,42,50),(7,51,43,17),(8,18,44,52),(9,53,45,19),(10,20,46,54),(11,55,47,21),(12,22,48,56),(25,86,83,72),(26,61,84,87),(27,88,73,62),(28,63,74,89),(29,90,75,64),(30,65,76,91),(31,92,77,66),(32,67,78,93),(33,94,79,68),(34,69,80,95),(35,96,81,70),(36,71,82,85)], [(1,86,37,72),(2,26,38,84),(3,88,39,62),(4,28,40,74),(5,90,41,64),(6,30,42,76),(7,92,43,66),(8,32,44,78),(9,94,45,68),(10,34,46,80),(11,96,47,70),(12,36,48,82),(13,73,59,27),(14,89,60,63),(15,75,49,29),(16,91,50,65),(17,77,51,31),(18,93,52,67),(19,79,53,33),(20,95,54,69),(21,81,55,35),(22,85,56,71),(23,83,57,25),(24,87,58,61)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,60),(23,59),(24,58),(25,88),(26,87),(27,86),(28,85),(29,96),(30,95),(31,94),(32,93),(33,92),(34,91),(35,90),(36,89),(37,39),(40,48),(41,47),(42,46),(43,45),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | SD16 | D12 | C8.C22 | S3xD4 | S3xD4 | S3xSD16 | Q16:S3 |
| kernel | Q8:3D12 | C6.SD16 | D6:C8 | C3xQ8:C4 | C12:D4 | C2xC24:C2 | C2xQ8:2S3 | C2xS3xQ8 | Q8:C4 | Dic6 | C2xDic3 | C3xQ8 | C22xS3 | C4:C4 | C2xC8 | C2xQ8 | D6 | Q8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Q8:3D12 ►in GL6(F73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 6 | 6 | 0 | 0 | 0 | 0 |
| 6 | 67 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[6,6,0,0,0,0,6,67,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,71,1],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,1,0,0,0,0,71,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1] >;
Q8:3D12 in GAP, Magma, Sage, TeX
Q_8\rtimes_3D_{12} % in TeX
G:=Group("Q8:3D12"); // GroupNames label
G:=SmallGroup(192,365);
// by ID
G=gap.SmallGroup(192,365);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,58,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=a*b,d*c*d=c^-1>;
// generators/relations