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## G = C6.Q16order 96 = 25·3

### 1st non-split extension by C6 of Q16 acting via Q16/Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C6.Q16
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×C3⋊C8 — C6.Q16
 Lower central C3 — C6 — C12 — C6.Q16
 Upper central C1 — C22 — C2×C4 — C4⋊C4

Generators and relations for C6.Q16
G = < a,b,c | a12=b4=1, c2=a9b2, bab-1=a7, cac-1=a5, cbc-1=a9b-1 >

Character table of C6.Q16

 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 0 0 0 0 0 0 -2 -2 2 √2 -√2 √2 -√2 0 0 0 0 0 0 orthogonal lifted from D8 ρ13 2 -2 2 -2 2 0 0 0 0 0 0 -2 -2 2 -√2 √2 -√2 √2 0 0 0 0 0 0 orthogonal lifted from D8 ρ14 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 ρ15 2 -2 -2 2 2 0 0 0 0 0 0 2 -2 -2 -√2 -√2 √2 √2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 ρ16 2 -2 -2 2 2 0 0 0 0 0 0 2 -2 -2 √2 √2 -√2 -√2 0 0 0 0 0 0 symplectic lifted from Q16, Schur index 2 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 symplectic lifted from Dic6, Schur index 2 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 symplectic lifted from Dic6, Schur index 2 ρ19 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 ρ20 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 ρ21 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 ρ22 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 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 D4⋊S3, Schur index 2 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 C3⋊Q16, Schur index 2

Smallest permutation representation of C6.Q16
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 71 58 27)(2 66 59 34)(3 61 60 29)(4 68 49 36)(5 63 50 31)(6 70 51 26)(7 65 52 33)(8 72 53 28)(9 67 54 35)(10 62 55 30)(11 69 56 25)(12 64 57 32)(13 46 92 74)(14 41 93 81)(15 48 94 76)(16 43 95 83)(17 38 96 78)(18 45 85 73)(19 40 86 80)(20 47 87 75)(21 42 88 82)(22 37 89 77)(23 44 90 84)(24 39 91 79)
(1 13 55 89 7 19 49 95)(2 18 56 94 8 24 50 88)(3 23 57 87 9 17 51 93)(4 16 58 92 10 22 52 86)(5 21 59 85 11 15 53 91)(6 14 60 90 12 20 54 96)(25 45 66 82 31 39 72 76)(26 38 67 75 32 44 61 81)(27 43 68 80 33 37 62 74)(28 48 69 73 34 42 63 79)(29 41 70 78 35 47 64 84)(30 46 71 83 36 40 65 77)```

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

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

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

Matrix representation of C6.Q16 in GL4(𝔽73) generated by

 0 1 0 0 72 72 0 0 0 0 72 2 0 0 72 1
,
 7 14 0 0 59 66 0 0 0 0 36 41 0 0 20 37
,
 60 30 0 0 43 13 0 0 0 0 32 41 0 0 16 0
`G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,72,72,0,0,2,1],[7,59,0,0,14,66,0,0,0,0,36,20,0,0,41,37],[60,43,0,0,30,13,0,0,0,0,32,16,0,0,41,0] >;`

C6.Q16 in GAP, Magma, Sage, TeX

`C_6.Q_{16}`
`% in TeX`

`G:=Group("C6.Q16");`
`// GroupNames label`

`G:=SmallGroup(96,14);`
`// by ID`

`G=gap.SmallGroup(96,14);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,121,31,297,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^4=1,c^2=a^9*b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;`
`// generators/relations`

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