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## G = C8.6D6order 96 = 25·3

### 3rd non-split extension by C8 of D6 acting via D6/S3=C2

Aliases: C8.6D6, C33SD32, Q161S3, C6.10D8, C12.5D4, D24.2C2, C24.4C22, C3⋊C163C2, (C3×Q16)⋊1C2, C2.6(D4⋊S3), C4.3(C3⋊D4), SmallGroup(96,35)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C8.6D6
 Chief series C1 — C3 — C6 — C12 — C24 — D24 — C8.6D6
 Lower central C3 — C6 — C12 — C24 — C8.6D6
 Upper central C1 — C2 — C4 — C8 — Q16

Generators and relations for C8.6D6
G = < a,b,c | a8=1, b6=a4, c2=a3, bab-1=a-1, ac=ca, cbc-1=a-1b5 >

Character table of C8.6D6

 class 1 2A 2B 3 4A 4B 6 8A 8B 12A 12B 12C 16A 16B 16C 16D 24A 24B size 1 1 24 2 2 8 2 2 2 4 8 8 6 6 6 6 4 4 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 0 -1 2 2 -1 2 2 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 -1 2 -2 -1 2 2 -1 1 1 0 0 0 0 -1 -1 orthogonal lifted from D6 ρ7 2 2 0 2 2 0 2 -2 -2 2 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ8 2 2 0 2 -2 0 2 0 0 -2 0 0 -√2 √2 -√2 √2 0 0 orthogonal lifted from D8 ρ9 2 2 0 2 -2 0 2 0 0 -2 0 0 √2 -√2 √2 -√2 0 0 orthogonal lifted from D8 ρ10 2 2 0 -1 2 0 -1 -2 -2 -1 √-3 -√-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ11 2 2 0 -1 2 0 -1 -2 -2 -1 -√-3 √-3 0 0 0 0 1 1 complex lifted from C3⋊D4 ρ12 2 -2 0 2 0 0 -2 √2 -√2 0 0 0 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 -√2 √2 complex lifted from SD32 ρ13 2 -2 0 2 0 0 -2 -√2 √2 0 0 0 ζ167+ζ16 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 √2 -√2 complex lifted from SD32 ρ14 2 -2 0 2 0 0 -2 -√2 √2 0 0 0 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 ζ165+ζ163 √2 -√2 complex lifted from SD32 ρ15 2 -2 0 2 0 0 -2 √2 -√2 0 0 0 ζ165+ζ163 ζ1615+ζ169 ζ1613+ζ1611 ζ167+ζ16 -√2 √2 complex lifted from SD32 ρ16 4 4 0 -2 -4 0 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4⋊S3, Schur index 2 ρ17 4 -4 0 -2 0 0 2 2√2 -2√2 0 0 0 0 0 0 0 √2 -√2 orthogonal faithful, Schur index 2 ρ18 4 -4 0 -2 0 0 2 -2√2 2√2 0 0 0 0 0 0 0 -√2 √2 orthogonal faithful, Schur index 2

Smallest permutation representation of C8.6D6
On 48 points
Generators in S48
```(1 7 13 3 9 15 5 11)(2 8 14 4 10 16 6 12)(17 23 29 19 25 31 21 27)(18 24 30 20 26 32 22 28)(33 39 45 35 41 47 37 43)(34 40 46 36 42 48 38 44)
(1 46 21 22 45 2 9 38 29 30 37 10)(3 44 23 20 47 16 11 36 31 28 39 8)(4 7 40 27 32 35 12 15 48 19 24 43)(5 42 25 18 33 14 13 34 17 26 41 6)
(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)```

`G:=sub<Sym(48)| (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44), (1,46,21,22,45,2,9,38,29,30,37,10)(3,44,23,20,47,16,11,36,31,28,39,8)(4,7,40,27,32,35,12,15,48,19,24,43)(5,42,25,18,33,14,13,34,17,26,41,6), (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)>;`

`G:=Group( (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44), (1,46,21,22,45,2,9,38,29,30,37,10)(3,44,23,20,47,16,11,36,31,28,39,8)(4,7,40,27,32,35,12,15,48,19,24,43)(5,42,25,18,33,14,13,34,17,26,41,6), (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) );`

`G=PermutationGroup([[(1,7,13,3,9,15,5,11),(2,8,14,4,10,16,6,12),(17,23,29,19,25,31,21,27),(18,24,30,20,26,32,22,28),(33,39,45,35,41,47,37,43),(34,40,46,36,42,48,38,44)], [(1,46,21,22,45,2,9,38,29,30,37,10),(3,44,23,20,47,16,11,36,31,28,39,8),(4,7,40,27,32,35,12,15,48,19,24,43),(5,42,25,18,33,14,13,34,17,26,41,6)], [(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)]])`

C8.6D6 is a maximal subgroup of
S3×SD32  D48⋊C2  Q32⋊S3  D485C2  C24.27C23  Q16⋊D6  Q16.D6  C9⋊SD32  D24.S3  C24.49D6  C3210SD32  C15⋊SD32  C24.D10  C8.6D30
C8.6D6 is a maximal quotient of
C6.SD32  C6.D16  C6.5Q32  C9⋊SD32  D24.S3  C24.49D6  C3210SD32  C15⋊SD32  C24.D10  C8.6D30

Matrix representation of C8.6D6 in GL4(𝔽7) generated by

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

C8.6D6 in GAP, Magma, Sage, TeX

`C_8._6D_6`
`% in TeX`

`G:=Group("C8.6D6");`
`// GroupNames label`

`G:=SmallGroup(96,35);`
`// by ID`

`G=gap.SmallGroup(96,35);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,73,103,218,116,122,579,297,69,2309]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^6=a^4,c^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^5>;`
`// generators/relations`

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