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## G = C12⋊Q8order 96 = 25·3

### The semidirect product of C12 and Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C12⋊Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C4×Dic3 — C12⋊Q8
 Lower central C3 — C2×C6 — C12⋊Q8
 Upper central C1 — C22 — C4⋊C4

Generators and relations for C12⋊Q8
G = < a,b,c | a12=b4=1, c2=b2, bab-1=a7, cac-1=a5, cbc-1=b-1 >

Subgroups: 138 in 68 conjugacy classes, 37 normal (19 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×8], C22, C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C42, C4⋊C4, C4⋊C4 [×3], C2×Q8 [×2], Dic6 [×4], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C4⋊Q8, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, C2×Dic6 [×2], C12⋊Q8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], C2×D4, C2×Q8 [×2], Dic6 [×2], C22×S3, C4⋊Q8, C2×Dic6, S3×D4, S3×Q8, C12⋊Q8

Character table of C12⋊Q8

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 12A 12B 12C 12D 12E 12F size 1 1 1 1 2 2 2 4 4 6 6 6 6 12 12 2 2 2 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 1 -1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ7 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ9 2 2 2 2 -1 -2 -2 -2 2 0 0 0 0 0 0 -1 -1 -1 1 1 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 2 2 2 -1 2 2 -2 -2 0 0 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 -1 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 -1 -2 -2 2 -2 0 0 0 0 0 0 -1 -1 -1 1 -1 1 1 1 -1 orthogonal lifted from D6 ρ13 2 -2 2 -2 2 0 0 0 0 0 -2 0 2 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 2 -2 2 0 0 0 0 0 2 0 -2 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 -2 2 -2 2 0 0 -2 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 -2 -2 2 2 0 0 0 0 -2 0 2 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 -2 -2 2 2 0 0 0 0 2 0 -2 0 0 0 -2 -2 2 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ18 2 2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 -2 2 -2 -2 0 0 2 0 0 symplectic lifted from Q8, Schur index 2 ρ19 2 2 -2 -2 -1 2 -2 0 0 0 0 0 0 0 0 1 -1 1 1 √3 -√3 -1 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ20 2 2 -2 -2 -1 2 -2 0 0 0 0 0 0 0 0 1 -1 1 1 -√3 √3 -1 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ21 2 2 -2 -2 -1 -2 2 0 0 0 0 0 0 0 0 1 -1 1 -1 √3 √3 1 -√3 -√3 symplectic lifted from Dic6, Schur index 2 ρ22 2 2 -2 -2 -1 -2 2 0 0 0 0 0 0 0 0 1 -1 1 -1 -√3 -√3 1 √3 √3 symplectic lifted from Dic6, Schur index 2 ρ23 4 -4 4 -4 -2 0 0 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 -4 -4 4 -2 0 0 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of C12⋊Q8
Regular action on 96 points
Generators in S96
```(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 86 73 25)(2 93 74 32)(3 88 75 27)(4 95 76 34)(5 90 77 29)(6 85 78 36)(7 92 79 31)(8 87 80 26)(9 94 81 33)(10 89 82 28)(11 96 83 35)(12 91 84 30)(13 58 63 46)(14 53 64 41)(15 60 65 48)(16 55 66 43)(17 50 67 38)(18 57 68 45)(19 52 69 40)(20 59 70 47)(21 54 71 42)(22 49 72 37)(23 56 61 44)(24 51 62 39)
(1 15 73 65)(2 20 74 70)(3 13 75 63)(4 18 76 68)(5 23 77 61)(6 16 78 66)(7 21 79 71)(8 14 80 64)(9 19 81 69)(10 24 82 62)(11 17 83 67)(12 22 84 72)(25 60 86 48)(26 53 87 41)(27 58 88 46)(28 51 89 39)(29 56 90 44)(30 49 91 37)(31 54 92 42)(32 59 93 47)(33 52 94 40)(34 57 95 45)(35 50 96 38)(36 55 85 43)```

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

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

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

Matrix representation of C12⋊Q8 in GL4(𝔽13) generated by

 10 0 0 0 2 4 0 0 0 0 7 11 0 0 12 6
,
 8 0 0 0 1 5 0 0 0 0 9 3 0 0 3 4
,
 3 4 0 0 4 10 0 0 0 0 6 2 0 0 1 7
`G:=sub<GL(4,GF(13))| [10,2,0,0,0,4,0,0,0,0,7,12,0,0,11,6],[8,1,0,0,0,5,0,0,0,0,9,3,0,0,3,4],[3,4,0,0,4,10,0,0,0,0,6,1,0,0,2,7] >;`

C12⋊Q8 in GAP, Magma, Sage, TeX

`C_{12}\rtimes Q_8`
`% in TeX`

`G:=Group("C12:Q8");`
`// GroupNames label`

`G:=SmallGroup(96,95);`
`// by ID`

`G=gap.SmallGroup(96,95);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,103,218,188,50,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^4=1,c^2=b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=b^-1>;`
`// generators/relations`

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