metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊4Q8, C4.13D24, C12.13D8, C42.37D6, C4⋊C8⋊4S3, C6.8(C2×D8), (C2×C8).22D6, C4.44(S3×Q8), C24⋊1C4⋊12C2, C3⋊2(D4⋊Q8), C2.10(C2×D24), C12⋊2Q8⋊13C2, (C4×D12).12C2, (C2×C4).134D12, (C2×C12).123D4, C12.103(C2×Q8), C2.D24.3C2, (C4×C12).72C22, (C2×C24).25C22, C6.31(C22⋊Q8), C12.287(C4○D4), (C2×C12).756C23, C2.12(C4.D12), C2.19(C8.D6), C22.119(C2×D12), C6.16(C8.C22), C4⋊Dic3.19C22, C4.111(D4⋊2S3), (C2×D12).196C22, (C3×C4⋊C8)⋊6C2, (C2×C6).139(C2×D4), (C2×C4).701(C22×S3), SmallGroup(192,405)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊4Q8
G = < a,b,c,d | a12=b2=c4=1, d2=c2, bab=cac-1=a-1, ad=da, cbc-1=a7b, bd=db, dcd-1=c-1 >
Subgroups: 360 in 108 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C24, Dic6, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, D4⋊Q8, C24⋊1C4, C2.D24, C3×C4⋊C8, C12⋊2Q8, C4×D12, D12⋊4Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, D8, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C2×D8, C8.C22, D24, C2×D12, D4⋊2S3, S3×Q8, D4⋊Q8, C4.D12, C2×D24, C8.D6, D12⋊4Q8
►On 96 points
Generators in S96
(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 77)(2 76)(3 75)(4 74)(5 73)(6 84)(7 83)(8 82)(9 81)(10 80)(11 79)(12 78)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 60)(24 59)(25 70)(26 69)(27 68)(28 67)(29 66)(30 65)(31 64)(32 63)(33 62)(34 61)(35 72)(36 71)(37 88)(38 87)(39 86)(40 85)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)
(1 28 78 71)(2 27 79 70)(3 26 80 69)(4 25 81 68)(5 36 82 67)(6 35 83 66)(7 34 84 65)(8 33 73 64)(9 32 74 63)(10 31 75 62)(11 30 76 61)(12 29 77 72)(13 46 57 85)(14 45 58 96)(15 44 59 95)(16 43 60 94)(17 42 49 93)(18 41 50 92)(19 40 51 91)(20 39 52 90)(21 38 53 89)(22 37 54 88)(23 48 55 87)(24 47 56 86)
(1 14 78 58)(2 15 79 59)(3 16 80 60)(4 17 81 49)(5 18 82 50)(6 19 83 51)(7 20 84 52)(8 21 73 53)(9 22 74 54)(10 23 75 55)(11 24 76 56)(12 13 77 57)(25 93 68 42)(26 94 69 43)(27 95 70 44)(28 96 71 45)(29 85 72 46)(30 86 61 47)(31 87 62 48)(32 88 63 37)(33 89 64 38)(34 90 65 39)(35 91 66 40)(36 92 67 41)
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,60)(24,59)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(37,88)(38,87)(39,86)(40,85)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89), (1,28,78,71)(2,27,79,70)(3,26,80,69)(4,25,81,68)(5,36,82,67)(6,35,83,66)(7,34,84,65)(8,33,73,64)(9,32,74,63)(10,31,75,62)(11,30,76,61)(12,29,77,72)(13,46,57,85)(14,45,58,96)(15,44,59,95)(16,43,60,94)(17,42,49,93)(18,41,50,92)(19,40,51,91)(20,39,52,90)(21,38,53,89)(22,37,54,88)(23,48,55,87)(24,47,56,86), (1,14,78,58)(2,15,79,59)(3,16,80,60)(4,17,81,49)(5,18,82,50)(6,19,83,51)(7,20,84,52)(8,21,73,53)(9,22,74,54)(10,23,75,55)(11,24,76,56)(12,13,77,57)(25,93,68,42)(26,94,69,43)(27,95,70,44)(28,96,71,45)(29,85,72,46)(30,86,61,47)(31,87,62,48)(32,88,63,37)(33,89,64,38)(34,90,65,39)(35,91,66,40)(36,92,67,41)>;
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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,60)(24,59)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,72)(36,71)(37,88)(38,87)(39,86)(40,85)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89), (1,28,78,71)(2,27,79,70)(3,26,80,69)(4,25,81,68)(5,36,82,67)(6,35,83,66)(7,34,84,65)(8,33,73,64)(9,32,74,63)(10,31,75,62)(11,30,76,61)(12,29,77,72)(13,46,57,85)(14,45,58,96)(15,44,59,95)(16,43,60,94)(17,42,49,93)(18,41,50,92)(19,40,51,91)(20,39,52,90)(21,38,53,89)(22,37,54,88)(23,48,55,87)(24,47,56,86), (1,14,78,58)(2,15,79,59)(3,16,80,60)(4,17,81,49)(5,18,82,50)(6,19,83,51)(7,20,84,52)(8,21,73,53)(9,22,74,54)(10,23,75,55)(11,24,76,56)(12,13,77,57)(25,93,68,42)(26,94,69,43)(27,95,70,44)(28,96,71,45)(29,85,72,46)(30,86,61,47)(31,87,62,48)(32,88,63,37)(33,89,64,38)(34,90,65,39)(35,91,66,40)(36,92,67,41) );
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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,84),(7,83),(8,82),(9,81),(10,80),(11,79),(12,78),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,50),(22,49),(23,60),(24,59),(25,70),(26,69),(27,68),(28,67),(29,66),(30,65),(31,64),(32,63),(33,62),(34,61),(35,72),(36,71),(37,88),(38,87),(39,86),(40,85),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89)], [(1,28,78,71),(2,27,79,70),(3,26,80,69),(4,25,81,68),(5,36,82,67),(6,35,83,66),(7,34,84,65),(8,33,73,64),(9,32,74,63),(10,31,75,62),(11,30,76,61),(12,29,77,72),(13,46,57,85),(14,45,58,96),(15,44,59,95),(16,43,60,94),(17,42,49,93),(18,41,50,92),(19,40,51,91),(20,39,52,90),(21,38,53,89),(22,37,54,88),(23,48,55,87),(24,47,56,86)], [(1,14,78,58),(2,15,79,59),(3,16,80,60),(4,17,81,49),(5,18,82,50),(6,19,83,51),(7,20,84,52),(8,21,73,53),(9,22,74,54),(10,23,75,55),(11,24,76,56),(12,13,77,57),(25,93,68,42),(26,94,69,43),(27,95,70,44),(28,96,71,45),(29,85,72,46),(30,86,61,47),(31,87,62,48),(32,88,63,37),(33,89,64,38),(34,90,65,39),(35,91,66,40),(36,92,67,41)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | D8 | C4○D4 | D12 | D24 | C8.C22 | D4⋊2S3 | S3×Q8 | C8.D6 |
| kernel | D12⋊4Q8 | C24⋊1C4 | C2.D24 | C3×C4⋊C8 | C12⋊2Q8 | C4×D12 | C4⋊C8 | D12 | C2×C12 | C42 | C2×C8 | C12 | C12 | C2×C4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 1 | 2 |
Matrix representation of D12⋊4Q8 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 7 | 66 |
| 0 | 0 | 7 | 14 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 7 | 66 |
| 0 | 0 | 59 | 66 |
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 18 | 68 |
| 0 | 0 | 50 | 55 |
| 27 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,7,7,0,0,66,14],[1,0,0,0,0,72,0,0,0,0,7,59,0,0,66,66],[0,72,0,0,1,0,0,0,0,0,18,50,0,0,68,55],[27,0,0,0,0,46,0,0,0,0,1,0,0,0,0,1] >;
D12⋊4Q8 in GAP, Magma, Sage, TeX
D_{12}\rtimes_4Q_8 % in TeX
G:=Group("D12:4Q8"); // GroupNames label
G:=SmallGroup(192,405);
// by ID
G=gap.SmallGroup(192,405);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,310,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^4=1,d^2=c^2,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^7*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations