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G = C12.Q8order 96 = 25·3

2nd non-split extension by C12 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.Q8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C12.Q8
 Lower central C3 — C6 — C12 — C12.Q8
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C12.Q8
G = < a,b,c | a4=b12=1, c2=ab6, bab-1=a-1, ac=ca, cbc-1=a-1b-1 >

Character table of C12.Q8

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F size 1 1 1 1 2 2 2 4 4 12 12 2 2 2 6 6 6 6 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 i -i -i i -1 1 -1 -1 1 1 -1 i -i -i 1 -1 i linear of order 4 ρ6 1 1 -1 -1 1 -1 1 -i i i -i -1 1 -1 -1 1 1 -1 -i i i 1 -1 -i linear of order 4 ρ7 1 1 -1 -1 1 -1 1 i -i i -i -1 1 -1 1 -1 -1 1 i -i -i 1 -1 i linear of order 4 ρ8 1 1 -1 -1 1 -1 1 -i i -i i -1 1 -1 1 -1 -1 1 -i i i 1 -1 -i linear of order 4 ρ9 2 2 2 2 -1 2 2 2 2 0 0 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 2 2 -1 2 2 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 -1 -1 1 orthogonal lifted from D6 ρ11 2 2 2 2 2 -2 -2 0 0 0 0 2 2 2 0 0 0 0 0 0 0 -2 -2 0 orthogonal lifted from D4 ρ12 2 2 -2 -2 2 2 -2 0 0 0 0 -2 2 -2 0 0 0 0 0 0 0 -2 2 0 symplectic lifted from Q8, Schur index 2 ρ13 2 2 -2 -2 -1 2 -2 0 0 0 0 1 -1 1 0 0 0 0 -√3 -√3 √3 1 -1 √3 symplectic lifted from Dic6, Schur index 2 ρ14 2 2 -2 -2 -1 2 -2 0 0 0 0 1 -1 1 0 0 0 0 √3 √3 -√3 1 -1 -√3 symplectic lifted from Dic6, Schur index 2 ρ15 2 2 -2 -2 -1 -2 2 2i -2i 0 0 1 -1 1 0 0 0 0 -i i i -1 1 -i complex lifted from C4×S3 ρ16 2 2 -2 -2 -1 -2 2 -2i 2i 0 0 1 -1 1 0 0 0 0 i -i -i -1 1 i complex lifted from C4×S3 ρ17 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 √-3 -√-3 √-3 1 1 -√-3 complex lifted from C3⋊D4 ρ18 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 0 0 0 0 -√-3 √-3 -√-3 1 1 √-3 complex lifted from C3⋊D4 ρ19 2 -2 -2 2 2 0 0 0 0 0 0 2 -2 -2 √-2 √-2 -√-2 -√-2 0 0 0 0 0 0 complex lifted from SD16 ρ20 2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 2 √-2 -√-2 √-2 -√-2 0 0 0 0 0 0 complex lifted from SD16 ρ21 2 -2 -2 2 2 0 0 0 0 0 0 2 -2 -2 -√-2 -√-2 √-2 √-2 0 0 0 0 0 0 complex lifted from SD16 ρ22 2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 2 -√-2 √-2 -√-2 √-2 0 0 0 0 0 0 complex lifted from SD16 ρ23 4 -4 -4 4 -2 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ24 4 -4 4 -4 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2

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

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

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

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

Matrix representation of C12.Q8 in GL5(𝔽73)

 72 0 0 0 0 0 72 2 0 0 0 72 1 0 0 0 0 0 72 0 0 0 0 0 72
,
 27 0 0 0 0 0 58 39 0 0 0 41 15 0 0 0 0 0 30 30 0 0 0 43 60
,
 72 0 0 0 0 0 0 12 0 0 0 67 12 0 0 0 0 0 5 54 0 0 0 59 68

`G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,72,0,0,0,2,1,0,0,0,0,0,72,0,0,0,0,0,72],[27,0,0,0,0,0,58,41,0,0,0,39,15,0,0,0,0,0,30,43,0,0,0,30,60],[72,0,0,0,0,0,0,67,0,0,0,12,12,0,0,0,0,0,5,59,0,0,0,54,68] >;`

C12.Q8 in GAP, Magma, Sage, TeX

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

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

`G:=SmallGroup(96,15);`
`// by ID`

`G=gap.SmallGroup(96,15);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,313,31,297,69,2309]);`
`// Polycyclic`

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

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