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## G = Q8.6D12order 192 = 26·3

### 1st non-split extension by Q8 of D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Q8.6D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C42⋊7S3 — Q8.6D12
 Lower central C3 — C6 — C2×C12 — Q8.6D12
 Upper central C1 — C22 — C42 — C4×Q8

Generators and relations for Q8.6D12
G = < a,b,c,d | a4=c12=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 328 in 112 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×6], C22, C22 [×3], S3, C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×2], Q8 [×3], C23, Dic3, C12 [×2], C12 [×5], D6 [×3], C2×C6, C42, C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×2], Q16 [×2], C2×D4, C2×Q8, C2×Q8, C3⋊C8 [×2], Dic6 [×2], D12 [×2], C2×Dic3, C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, C22×S3, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C2×C3⋊C8 [×2], D6⋊C4 [×2], Q82S3 [×2], C3⋊Q16 [×2], C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C2×D12, C6×Q8, Q8.D4, C12⋊C8, C6.D8, C6.SD16, C427S3, C2×Q82S3, C2×C3⋊Q16, Q8×C12, Q8.6D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C4○D8, C8.C22, C2×D12, C4○D12, C2×C3⋊D4, Q8.D4, C127D4, Q8.11D6, Q8.13D6, Q8.6D12

Smallest permutation representation of Q8.6D12
On 96 points
Generators in S96
```(1 28 37 71)(2 29 38 72)(3 30 39 61)(4 31 40 62)(5 32 41 63)(6 33 42 64)(7 34 43 65)(8 35 44 66)(9 36 45 67)(10 25 46 68)(11 26 47 69)(12 27 48 70)(13 52 74 95)(14 53 75 96)(15 54 76 85)(16 55 77 86)(17 56 78 87)(18 57 79 88)(19 58 80 89)(20 59 81 90)(21 60 82 91)(22 49 83 92)(23 50 84 93)(24 51 73 94)
(1 88 37 57)(2 89 38 58)(3 90 39 59)(4 91 40 60)(5 92 41 49)(6 93 42 50)(7 94 43 51)(8 95 44 52)(9 96 45 53)(10 85 46 54)(11 86 47 55)(12 87 48 56)(13 35 74 66)(14 36 75 67)(15 25 76 68)(16 26 77 69)(17 27 78 70)(18 28 79 71)(19 29 80 72)(20 30 81 61)(21 31 82 62)(22 32 83 63)(23 33 84 64)(24 34 73 65)
(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 48 37 12)(2 11 38 47)(3 46 39 10)(4 9 40 45)(5 44 41 8)(6 7 42 43)(13 92 74 49)(14 60 75 91)(15 90 76 59)(16 58 77 89)(17 88 78 57)(18 56 79 87)(19 86 80 55)(20 54 81 85)(21 96 82 53)(22 52 83 95)(23 94 84 51)(24 50 73 93)(25 61 68 30)(26 29 69 72)(27 71 70 28)(31 67 62 36)(32 35 63 66)(33 65 64 34)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E ··· 4I 4J 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12P order 1 2 2 2 2 3 4 4 4 4 4 ··· 4 4 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 size 1 1 1 1 24 2 2 2 2 2 4 ··· 4 24 2 2 2 12 12 12 12 2 2 2 2 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 D12 C4○D8 C4○D12 C8.C22 Q8.11D6 Q8.13D6 kernel Q8.6D12 C12⋊C8 C6.D8 C6.SD16 C42⋊7S3 C2×Q8⋊2S3 C2×C3⋊Q16 Q8×C12 C4×Q8 C2×C12 C3×Q8 C42 C4⋊C4 C2×Q8 C12 C2×C4 Q8 C6 C4 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 4 4 4 4 1 2 2

Matrix representation of Q8.6D12 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 1 3 0 0 48 72
,
 43 13 0 0 60 30 0 0 0 0 12 18 0 0 69 61
,
 7 66 0 0 7 14 0 0 0 0 46 0 0 0 0 46
,
 7 66 0 0 59 66 0 0 0 0 46 0 0 0 18 27
`G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,48,0,0,3,72],[43,60,0,0,13,30,0,0,0,0,12,69,0,0,18,61],[7,7,0,0,66,14,0,0,0,0,46,0,0,0,0,46],[7,59,0,0,66,66,0,0,0,0,46,18,0,0,0,27] >;`

Q8.6D12 in GAP, Magma, Sage, TeX

`Q_8._6D_{12}`
`% in TeX`

`G:=Group("Q8.6D12");`
`// GroupNames label`

`G:=SmallGroup(192,587);`
`// by ID`

`G=gap.SmallGroup(192,587);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,184,1123,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=c^12=1,b^2=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^-1*b,d*c*d^-1=a^2*c^-1>;`
`// generators/relations`

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