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## G = C12.6Q8order 96 = 25·3

### 3rd non-split extension by C12 of Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C12.6Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — Dic3⋊C4 — C12.6Q8
 Lower central C3 — C2×C6 — C12.6Q8
 Upper central C1 — C22 — C42

Generators and relations for C12.6Q8
G = < a,b,c | a12=b4=1, c2=a6b2, ab=ba, cac-1=a-1, cbc-1=a6b-1 >

Subgroups: 106 in 56 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, C2×C4, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×Dic3, C2×C12, C2×C12, C42.C2, Dic3⋊C4, C4⋊Dic3, C4×C12, C12.6Q8
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, Dic6, C22×S3, C42.C2, C2×Dic6, C4○D12, C12.6Q8

Character table of C12.6Q8

 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 12G 12H 12I 12J 12K 12L size 1 1 1 1 2 2 2 2 2 2 2 12 12 12 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 -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 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 -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 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 -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 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 -1 -1 1 -1 -1 1 linear of order 2 ρ9 2 2 2 2 -1 2 -2 -2 -2 2 -2 0 0 0 0 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 2 2 2 -1 2 2 2 2 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 2 -1 -2 -2 2 2 -2 -2 0 0 0 0 -1 -1 -1 1 -1 1 1 -1 1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ12 2 2 2 2 -1 -2 2 -2 -2 -2 2 0 0 0 0 -1 -1 -1 1 1 -1 1 1 -1 1 1 1 -1 -1 1 orthogonal lifted from D6 ρ13 2 -2 -2 2 2 2 0 0 0 -2 0 0 0 0 0 2 -2 -2 -2 0 0 -2 0 0 2 2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 -2 -2 2 2 -2 0 0 0 2 0 0 0 0 0 2 -2 -2 2 0 0 2 0 0 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 2 -2 -2 2 -1 -2 0 0 0 2 0 0 0 0 0 -1 1 1 -1 -√3 √3 -1 -√3 -√3 1 1 √3 -√3 √3 √3 symplectic lifted from Dic6, Schur index 2 ρ16 2 -2 -2 2 -1 -2 0 0 0 2 0 0 0 0 0 -1 1 1 -1 √3 -√3 -1 √3 √3 1 1 -√3 √3 -√3 -√3 symplectic lifted from Dic6, Schur index 2 ρ17 2 -2 -2 2 -1 2 0 0 0 -2 0 0 0 0 0 -1 1 1 1 -√3 -√3 1 -√3 √3 -1 -1 √3 √3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ18 2 -2 -2 2 -1 2 0 0 0 -2 0 0 0 0 0 -1 1 1 1 √3 √3 1 √3 -√3 -1 -1 -√3 -√3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ19 2 2 -2 -2 2 0 0 -2i 2i 0 0 0 0 0 0 -2 -2 2 0 2i 0 0 -2i 0 0 0 -2i 0 0 2i complex lifted from C4○D4 ρ20 2 -2 2 -2 2 0 -2i 0 0 0 2i 0 0 0 0 -2 2 -2 0 0 2i 0 0 2i 0 0 0 -2i -2i 0 complex lifted from C4○D4 ρ21 2 -2 2 -2 2 0 2i 0 0 0 -2i 0 0 0 0 -2 2 -2 0 0 -2i 0 0 -2i 0 0 0 2i 2i 0 complex lifted from C4○D4 ρ22 2 2 -2 -2 2 0 0 2i -2i 0 0 0 0 0 0 -2 -2 2 0 -2i 0 0 2i 0 0 0 2i 0 0 -2i complex lifted from C4○D4 ρ23 2 2 -2 -2 -1 0 0 2i -2i 0 0 0 0 0 0 1 1 -1 -√3 i -√-3 √3 -i √-3 √3 -√3 -i -√-3 √-3 i complex lifted from C4○D12 ρ24 2 -2 2 -2 -1 0 -2i 0 0 0 2i 0 0 0 0 1 -1 1 √3 -√-3 -i -√3 √-3 -i √3 -√3 -√-3 i i √-3 complex lifted from C4○D12 ρ25 2 -2 2 -2 -1 0 -2i 0 0 0 2i 0 0 0 0 1 -1 1 -√3 √-3 -i √3 -√-3 -i -√3 √3 √-3 i i -√-3 complex lifted from C4○D12 ρ26 2 2 -2 -2 -1 0 0 -2i 2i 0 0 0 0 0 0 1 1 -1 -√3 -i √-3 √3 i -√-3 √3 -√3 i √-3 -√-3 -i complex lifted from C4○D12 ρ27 2 2 -2 -2 -1 0 0 2i -2i 0 0 0 0 0 0 1 1 -1 √3 i √-3 -√3 -i -√-3 -√3 √3 -i √-3 -√-3 i complex lifted from C4○D12 ρ28 2 -2 2 -2 -1 0 2i 0 0 0 -2i 0 0 0 0 1 -1 1 √3 √-3 i -√3 -√-3 i √3 -√3 √-3 -i -i -√-3 complex lifted from C4○D12 ρ29 2 2 -2 -2 -1 0 0 -2i 2i 0 0 0 0 0 0 1 1 -1 √3 -i -√-3 -√3 i √-3 -√3 √3 i -√-3 √-3 -i complex lifted from C4○D12 ρ30 2 -2 2 -2 -1 0 2i 0 0 0 -2i 0 0 0 0 1 -1 1 -√3 -√-3 i √3 √-3 i -√3 √3 -√-3 -i -i √-3 complex lifted from C4○D12

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

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

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

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

Matrix representation of C12.6Q8 in GL4(𝔽13) generated by

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

C12.6Q8 in GAP, Magma, Sage, TeX

`C_{12}._6Q_8`
`% in TeX`

`G:=Group("C12.6Q8");`
`// GroupNames label`

`G:=SmallGroup(96,77);`
`// by ID`

`G=gap.SmallGroup(96,77);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,96,217,55,218,86,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^4=1,c^2=a^6*b^2,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;`
`// generators/relations`

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