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## G = Q8⋊2D12order 192 = 26·3

### The semidirect product of Q8 and D12 acting via D12/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — Q8⋊2D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4⋊D12 — Q8⋊2D12
 Lower central C3 — C6 — C2×C12 — Q8⋊2D12
 Upper central C1 — C22 — C42 — C4×Q8

Generators and relations for Q82D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >

Subgroups: 456 in 128 conjugacy classes, 47 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6 [×3], C8 [×2], C2×C4 [×3], C2×C4 [×2], D4 [×8], Q8 [×2], Q8, C23 [×2], C12 [×2], C12 [×2], C12 [×4], D6 [×6], C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C2×D4 [×4], C2×Q8, C3⋊C8 [×2], D12 [×8], C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, C22×S3 [×2], D4⋊C4 [×2], C4⋊C8, C4×Q8, C41D4, C2×SD16 [×2], C2×C3⋊C8 [×2], Q82S3 [×4], C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×D12 [×2], C2×D12 [×2], C6×Q8, C4⋊SD16, C12⋊C8, C6.D8 [×2], C4⋊D12, C2×Q82S3 [×2], Q8×C12, Q82D12
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], SD16 [×2], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, C2×SD16, C8⋊C22, Q82S3 [×2], C2×D12, C4○D12, C2×C3⋊D4, C4⋊SD16, C127D4, C2×Q82S3, D4⋊D6, Q82D12

Smallest permutation representation of Q82D12
On 96 points
Generators in S96
```(1 30 63 51)(2 31 64 52)(3 32 65 53)(4 33 66 54)(5 34 67 55)(6 35 68 56)(7 36 69 57)(8 25 70 58)(9 26 71 59)(10 27 72 60)(11 28 61 49)(12 29 62 50)(13 95 73 39)(14 96 74 40)(15 85 75 41)(16 86 76 42)(17 87 77 43)(18 88 78 44)(19 89 79 45)(20 90 80 46)(21 91 81 47)(22 92 82 48)(23 93 83 37)(24 94 84 38)
(1 17 63 77)(2 18 64 78)(3 19 65 79)(4 20 66 80)(5 21 67 81)(6 22 68 82)(7 23 69 83)(8 24 70 84)(9 13 71 73)(10 14 72 74)(11 15 61 75)(12 16 62 76)(25 38 58 94)(26 39 59 95)(27 40 60 96)(28 41 49 85)(29 42 50 86)(30 43 51 87)(31 44 52 88)(32 45 53 89)(33 46 54 90)(34 47 55 91)(35 48 56 92)(36 37 57 93)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 90)(14 89)(15 88)(16 87)(17 86)(18 85)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 55)(26 54)(27 53)(28 52)(29 51)(30 50)(31 49)(32 60)(33 59)(34 58)(35 57)(36 56)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 84)(48 83)(61 64)(62 63)(65 72)(66 71)(67 70)(68 69)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 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 S3 D4 D4 D6 D6 D6 SD16 C4○D4 C3⋊D4 D12 C4○D12 C8⋊C22 Q8⋊2S3 D4⋊D6 kernel Q8⋊2D12 C12⋊C8 C6.D8 C4⋊D12 C2×Q8⋊2S3 Q8×C12 C4×Q8 C2×C12 C3×Q8 C42 C4⋊C4 C2×Q8 C12 C12 C2×C4 Q8 C4 C6 C4 C2 # reps 1 1 2 1 2 1 1 2 2 1 1 1 4 2 4 4 4 1 2 2

Matrix representation of Q82D12 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 72 3 0 0 48 1
,
 72 0 0 0 0 72 0 0 0 0 0 18 0 0 4 0
,
 7 66 0 0 7 14 0 0 0 0 72 0 0 0 0 72
,
 7 66 0 0 59 66 0 0 0 0 72 0 0 0 48 1
`G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,48,0,0,3,1],[72,0,0,0,0,72,0,0,0,0,0,4,0,0,18,0],[7,7,0,0,66,14,0,0,0,0,72,0,0,0,0,72],[7,59,0,0,66,66,0,0,0,0,72,48,0,0,0,1] >;`

Q82D12 in GAP, Magma, Sage, TeX

`Q_8\rtimes_2D_{12}`
`% in TeX`

`G:=Group("Q8:2D12");`
`// GroupNames label`

`G:=SmallGroup(192,586);`
`// by ID`

`G=gap.SmallGroup(192,586);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,184,1123,297,136,6278]);`
`// 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,d*b*d=a^-1*b,d*c*d=c^-1>;`
`// generators/relations`

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