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## G = Q8⋊6D12order 192 = 26·3

### 1st semidirect product of Q8 and D12 acting through Inn(Q8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Q8⋊6D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — Q8⋊6D12
 Lower central C3 — C2×C6 — Q8⋊6D12
 Upper central C1 — C22 — C4×Q8

Generators and relations for Q86D12
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 [×3], C2 [×5], C3, C4 [×6], C4 [×8], C22, C22 [×13], S3 [×5], C6 [×3], C2×C4, C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×4], Q8 [×6], C23 [×4], Dic3 [×4], C12 [×6], C12 [×4], D6 [×2], D6 [×11], C2×C6, C42 [×3], C22⋊C4 [×10], C4⋊C4 [×3], C4⋊C4 [×3], C22×C4 [×6], C2×D4 [×6], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic6 [×6], C4×S3 [×12], D12 [×12], C2×Dic3, C2×Dic3 [×3], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C22×S3 [×3], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, C4⋊Dic3 [×3], D6⋊C4, D6⋊C4 [×9], C4×C12 [×3], C3×C4⋊C4 [×3], C2×Dic6 [×3], S3×C2×C4 [×6], C2×D12 [×6], S3×Q8 [×4], Q83S3 [×4], C6×Q8, Q85D4, C4×D12 [×3], C427S3 [×3], C12⋊D4 [×3], C4.D12 [×3], Q8×C12, C2×S3×Q8, C2×Q83S3, Q86D12
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], S3×C23, Q85D4, C22×D12, Q8.15D6, S3×C4○D4, Q86D12

Smallest permutation representation of Q86D12
On 96 points
Generators in S96
```(1 38 84 87)(2 39 73 88)(3 40 74 89)(4 41 75 90)(5 42 76 91)(6 43 77 92)(7 44 78 93)(8 45 79 94)(9 46 80 95)(10 47 81 96)(11 48 82 85)(12 37 83 86)(13 65 55 27)(14 66 56 28)(15 67 57 29)(16 68 58 30)(17 69 59 31)(18 70 60 32)(19 71 49 33)(20 72 50 34)(21 61 51 35)(22 62 52 36)(23 63 53 25)(24 64 54 26)
(1 18 84 60)(2 49 73 19)(3 20 74 50)(4 51 75 21)(5 22 76 52)(6 53 77 23)(7 24 78 54)(8 55 79 13)(9 14 80 56)(10 57 81 15)(11 16 82 58)(12 59 83 17)(25 43 63 92)(26 93 64 44)(27 45 65 94)(28 95 66 46)(29 47 67 96)(30 85 68 48)(31 37 69 86)(32 87 70 38)(33 39 71 88)(34 89 72 40)(35 41 61 90)(36 91 62 42)
(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 49)(14 60)(15 59)(16 58)(17 57)(18 56)(19 55)(20 54)(21 53)(22 52)(23 51)(24 50)(25 61)(26 72)(27 71)(28 70)(29 69)(30 68)(31 67)(32 66)(33 65)(34 64)(35 63)(36 62)(37 47)(38 46)(39 45)(40 44)(41 43)(73 79)(74 78)(75 77)(80 84)(81 83)(86 96)(87 95)(88 94)(89 93)(90 92)```

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

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

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

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 C4○D4 D12 2- 1+4 Q8.15D6 S3×C4○D4 kernel Q8⋊6D12 C4×D12 C42⋊7S3 C12⋊D4 C4.D12 Q8×C12 C2×S3×Q8 C2×Q8⋊3S3 C4×Q8 C3×Q8 C42 C4⋊C4 C2×Q8 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 Q86D12 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] >;`

Q86D12 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`

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