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## G = Q8⋊7D12order 192 = 26·3

### 2nd semidirect product of Q8 and D12 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Q8⋊7D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×Q8⋊3S3 — Q8⋊7D12
 Lower central C3 — C2×C6 — Q8⋊7D12
 Upper central C1 — C22 — C4×Q8

Generators and relations for Q87D12
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 [×3], C2 [×6], C3, C4 [×8], C4 [×5], C22, C22 [×18], S3 [×6], C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×14], D4 [×24], Q8 [×4], C23 [×6], Dic3 [×2], C12 [×8], C12 [×3], D6 [×18], C2×C6, C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4, C22×C4 [×6], C2×D4 [×15], C2×Q8, C4○D4 [×8], C4×S3 [×12], D12 [×24], C2×Dic3 [×2], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3 [×6], C4×D4 [×3], C4×Q8, C4⋊D4 [×6], C41D4 [×3], C2×C4○D4 [×2], C4⋊Dic3, D6⋊C4 [×6], C4×C12 [×3], C3×C4⋊C4 [×3], S3×C2×C4 [×6], C2×D12 [×15], Q83S3 [×8], C6×Q8, Q86D4, C4×D12 [×3], C4⋊D12 [×3], C12⋊D4 [×6], Q8×C12, C2×Q83S3 [×2], Q87D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C2×D12 [×6], Q83S3 [×2], S3×C23, Q86D4, C22×D12, C2×Q83S3, D4○D12, Q87D12

Smallest permutation representation of Q87D12
On 96 points
Generators in S96
```(1 35 42 50)(2 36 43 51)(3 25 44 52)(4 26 45 53)(5 27 46 54)(6 28 47 55)(7 29 48 56)(8 30 37 57)(9 31 38 58)(10 32 39 59)(11 33 40 60)(12 34 41 49)(13 92 62 74)(14 93 63 75)(15 94 64 76)(16 95 65 77)(17 96 66 78)(18 85 67 79)(19 86 68 80)(20 87 69 81)(21 88 70 82)(22 89 71 83)(23 90 72 84)(24 91 61 73)
(1 70 42 21)(2 71 43 22)(3 72 44 23)(4 61 45 24)(5 62 46 13)(6 63 47 14)(7 64 48 15)(8 65 37 16)(9 66 38 17)(10 67 39 18)(11 68 40 19)(12 69 41 20)(25 90 52 84)(26 91 53 73)(27 92 54 74)(28 93 55 75)(29 94 56 76)(30 95 57 77)(31 96 58 78)(32 85 59 79)(33 86 60 80)(34 87 49 81)(35 88 50 82)(36 89 51 83)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 60)(9 59)(10 58)(11 57)(12 56)(13 89)(14 88)(15 87)(16 86)(17 85)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 90)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)(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)| (1,35,42,50)(2,36,43,51)(3,25,44,52)(4,26,45,53)(5,27,46,54)(6,28,47,55)(7,29,48,56)(8,30,37,57)(9,31,38,58)(10,32,39,59)(11,33,40,60)(12,34,41,49)(13,92,62,74)(14,93,63,75)(15,94,64,76)(16,95,65,77)(17,96,66,78)(18,85,67,79)(19,86,68,80)(20,87,69,81)(21,88,70,82)(22,89,71,83)(23,90,72,84)(24,91,61,73), (1,70,42,21)(2,71,43,22)(3,72,44,23)(4,61,45,24)(5,62,46,13)(6,63,47,14)(7,64,48,15)(8,65,37,16)(9,66,38,17)(10,67,39,18)(11,68,40,19)(12,69,41,20)(25,90,52,84)(26,91,53,73)(27,92,54,74)(28,93,55,75)(29,94,56,76)(30,95,57,77)(31,96,58,78)(32,85,59,79)(33,86,60,80)(34,87,49,81)(35,88,50,82)(36,89,51,83), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,60)(9,59)(10,58)(11,57)(12,56)(13,89)(14,88)(15,87)(16,86)(17,85)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46)(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( (1,35,42,50)(2,36,43,51)(3,25,44,52)(4,26,45,53)(5,27,46,54)(6,28,47,55)(7,29,48,56)(8,30,37,57)(9,31,38,58)(10,32,39,59)(11,33,40,60)(12,34,41,49)(13,92,62,74)(14,93,63,75)(15,94,64,76)(16,95,65,77)(17,96,66,78)(18,85,67,79)(19,86,68,80)(20,87,69,81)(21,88,70,82)(22,89,71,83)(23,90,72,84)(24,91,61,73), (1,70,42,21)(2,71,43,22)(3,72,44,23)(4,61,45,24)(5,62,46,13)(6,63,47,14)(7,64,48,15)(8,65,37,16)(9,66,38,17)(10,67,39,18)(11,68,40,19)(12,69,41,20)(25,90,52,84)(26,91,53,73)(27,92,54,74)(28,93,55,75)(29,94,56,76)(30,95,57,77)(31,96,58,78)(32,85,59,79)(33,86,60,80)(34,87,49,81)(35,88,50,82)(36,89,51,83), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,60)(9,59)(10,58)(11,57)(12,56)(13,89)(14,88)(15,87)(16,86)(17,85)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,90)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46)(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([(1,35,42,50),(2,36,43,51),(3,25,44,52),(4,26,45,53),(5,27,46,54),(6,28,47,55),(7,29,48,56),(8,30,37,57),(9,31,38,58),(10,32,39,59),(11,33,40,60),(12,34,41,49),(13,92,62,74),(14,93,63,75),(15,94,64,76),(16,95,65,77),(17,96,66,78),(18,85,67,79),(19,86,68,80),(20,87,69,81),(21,88,70,82),(22,89,71,83),(23,90,72,84),(24,91,61,73)], [(1,70,42,21),(2,71,43,22),(3,72,44,23),(4,61,45,24),(5,62,46,13),(6,63,47,14),(7,64,48,15),(8,65,37,16),(9,66,38,17),(10,67,39,18),(11,68,40,19),(12,69,41,20),(25,90,52,84),(26,91,53,73),(27,92,54,74),(28,93,55,75),(29,94,56,76),(30,95,57,77),(31,96,58,78),(32,85,59,79),(33,86,60,80),(34,87,49,81),(35,88,50,82),(36,89,51,83)], [(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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,60),(9,59),(10,58),(11,57),(12,56),(13,89),(14,88),(15,87),(16,86),(17,85),(18,96),(19,95),(20,94),(21,93),(22,92),(23,91),(24,90),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,48),(35,47),(36,46),(61,84),(62,83),(63,82),(64,81),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75),(71,74),(72,73)])`

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 Q87D12 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] >;`

Q87D12 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`

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