metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊7D12, C42.129D6, C6.1102+ 1+4, (C3×Q8)⋊11D4, (C4×Q8)⋊14S3, (C4×D12)⋊39C2, C4⋊C4.296D6, (Q8×C12)⋊12C2, C3⋊2(Q8⋊6D4), C12.58(C2×D4), C4.26(C2×D12), C12⋊D4⋊18C2, C12⋊17(C4○D4), C4⋊D12⋊13C2, C4⋊3(Q8⋊3S3), (C2×Q8).229D6, C6.20(C22×D4), C2.22(D4○D12), (C2×C6).121C24, C2.22(C22×D12), (C2×C12).170C23, (C4×C12).173C22, D6⋊C4.101C22, (C2×D12).28C22, (C6×Q8).221C22, (C22×S3).46C23, C4⋊Dic3.399C22, C22.142(S3×C23), (C2×Dic3).215C23, (C2×Q8⋊3S3)⋊4C2, C6.112(C2×C4○D4), (S3×C2×C4).73C22, C2.11(C2×Q8⋊3S3), (C3×C4⋊C4).349C22, (C2×C4).734(C22×S3), SmallGroup(192,1136)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊7D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >
Subgroups: 936 in 312 conjugacy classes, 115 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4×D4, C4×Q8, C4⋊D4, C4⋊1D4, C2×C4○D4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, S3×C2×C4, C2×D12, Q8⋊3S3, C6×Q8, Q8⋊6D4, C4×D12, C4⋊D12, C12⋊D4, Q8×C12, C2×Q8⋊3S3, Q8⋊7D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, D12, C22×S3, C22×D4, C2×C4○D4, 2+ 1+4, C2×D12, Q8⋊3S3, S3×C23, Q8⋊6D4, C22×D12, C2×Q8⋊3S3, D4○D12, Q8⋊7D12
►On 96 points
(1 32 80 64)(2 33 81 65)(3 34 82 66)(4 35 83 67)(5 36 84 68)(6 25 73 69)(7 26 74 70)(8 27 75 71)(9 28 76 72)(10 29 77 61)(11 30 78 62)(12 31 79 63)(13 44 54 95)(14 45 55 96)(15 46 56 85)(16 47 57 86)(17 48 58 87)(18 37 59 88)(19 38 60 89)(20 39 49 90)(21 40 50 91)(22 41 51 92)(23 42 52 93)(24 43 53 94)
(1 22 80 51)(2 23 81 52)(3 24 82 53)(4 13 83 54)(5 14 84 55)(6 15 73 56)(7 16 74 57)(8 17 75 58)(9 18 76 59)(10 19 77 60)(11 20 78 49)(12 21 79 50)(25 85 69 46)(26 86 70 47)(27 87 71 48)(28 88 72 37)(29 89 61 38)(30 90 62 39)(31 91 63 40)(32 92 64 41)(33 93 65 42)(34 94 66 43)(35 95 67 44)(36 96 68 45)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 72)(11 71)(12 70)(13 43)(14 42)(15 41)(16 40)(17 39)(18 38)(19 37)(20 48)(21 47)(22 46)(23 45)(24 44)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 84)(34 83)(35 82)(36 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)
G:=sub<Sym(96)| (1,32,80,64)(2,33,81,65)(3,34,82,66)(4,35,83,67)(5,36,84,68)(6,25,73,69)(7,26,74,70)(8,27,75,71)(9,28,76,72)(10,29,77,61)(11,30,78,62)(12,31,79,63)(13,44,54,95)(14,45,55,96)(15,46,56,85)(16,47,57,86)(17,48,58,87)(18,37,59,88)(19,38,60,89)(20,39,49,90)(21,40,50,91)(22,41,51,92)(23,42,52,93)(24,43,53,94), (1,22,80,51)(2,23,81,52)(3,24,82,53)(4,13,83,54)(5,14,84,55)(6,15,73,56)(7,16,74,57)(8,17,75,58)(9,18,76,59)(10,19,77,60)(11,20,78,49)(12,21,79,50)(25,85,69,46)(26,86,70,47)(27,87,71,48)(28,88,72,37)(29,89,61,38)(30,90,62,39)(31,91,63,40)(32,92,64,41)(33,93,65,42)(34,94,66,43)(35,95,67,44)(36,96,68,45), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,72)(11,71)(12,70)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,48)(21,47)(22,46)(23,45)(24,44)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,84)(34,83)(35,82)(36,81)(49,87)(50,86)(51,85)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)>;
G:=Group( (1,32,80,64)(2,33,81,65)(3,34,82,66)(4,35,83,67)(5,36,84,68)(6,25,73,69)(7,26,74,70)(8,27,75,71)(9,28,76,72)(10,29,77,61)(11,30,78,62)(12,31,79,63)(13,44,54,95)(14,45,55,96)(15,46,56,85)(16,47,57,86)(17,48,58,87)(18,37,59,88)(19,38,60,89)(20,39,49,90)(21,40,50,91)(22,41,51,92)(23,42,52,93)(24,43,53,94), (1,22,80,51)(2,23,81,52)(3,24,82,53)(4,13,83,54)(5,14,84,55)(6,15,73,56)(7,16,74,57)(8,17,75,58)(9,18,76,59)(10,19,77,60)(11,20,78,49)(12,21,79,50)(25,85,69,46)(26,86,70,47)(27,87,71,48)(28,88,72,37)(29,89,61,38)(30,90,62,39)(31,91,63,40)(32,92,64,41)(33,93,65,42)(34,94,66,43)(35,95,67,44)(36,96,68,45), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,72)(11,71)(12,70)(13,43)(14,42)(15,41)(16,40)(17,39)(18,38)(19,37)(20,48)(21,47)(22,46)(23,45)(24,44)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,84)(34,83)(35,82)(36,81)(49,87)(50,86)(51,85)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88) );
G=PermutationGroup([[(1,32,80,64),(2,33,81,65),(3,34,82,66),(4,35,83,67),(5,36,84,68),(6,25,73,69),(7,26,74,70),(8,27,75,71),(9,28,76,72),(10,29,77,61),(11,30,78,62),(12,31,79,63),(13,44,54,95),(14,45,55,96),(15,46,56,85),(16,47,57,86),(17,48,58,87),(18,37,59,88),(19,38,60,89),(20,39,49,90),(21,40,50,91),(22,41,51,92),(23,42,52,93),(24,43,53,94)], [(1,22,80,51),(2,23,81,52),(3,24,82,53),(4,13,83,54),(5,14,84,55),(6,15,73,56),(7,16,74,57),(8,17,75,58),(9,18,76,59),(10,19,77,60),(11,20,78,49),(12,21,79,50),(25,85,69,46),(26,86,70,47),(27,87,71,48),(28,88,72,37),(29,89,61,38),(30,90,62,39),(31,91,63,40),(32,92,64,41),(33,93,65,42),(34,94,66,43),(35,95,67,44),(36,96,68,45)], [(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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,72),(11,71),(12,70),(13,43),(14,42),(15,41),(16,40),(17,39),(18,38),(19,37),(20,48),(21,47),(22,46),(23,45),(24,44),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,84),(34,83),(35,82),(36,81),(49,87),(50,86),(51,85),(52,96),(53,95),(54,94),(55,93),(56,92),(57,91),(58,90),(59,89),(60,88)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 12 | ··· | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | D12 | 2+ 1+4 | Q8⋊3S3 | D4○D12 |
| kernel | Q8⋊7D12 | C4×D12 | C4⋊D12 | C12⋊D4 | Q8×C12 | C2×Q8⋊3S3 | C4×Q8 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | Q8 | C6 | C4 | C2 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 1 | 4 | 3 | 3 | 1 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of Q8⋊7D12 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 8 |
| 0 | 0 | 3 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 12 |
| 0 | 0 | 0 | 8 |
| 3 | 10 | 0 | 0 |
| 3 | 6 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 10 | 3 | 0 | 0 |
| 6 | 3 | 0 | 0 |
| 0 | 0 | 1 | 5 |
| 0 | 0 | 0 | 12 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,12,3,0,0,8,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,12,8],[3,3,0,0,10,6,0,0,0,0,12,0,0,0,0,12],[10,6,0,0,3,3,0,0,0,0,1,0,0,0,5,12] >;
Q8⋊7D12 in GAP, Magma, Sage, TeX
Q_8\rtimes_7D_{12} % in TeX
G:=Group("Q8:7D12"); // GroupNames label
G:=SmallGroup(192,1136);
// by ID
G=gap.SmallGroup(192,1136);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,184,675,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations