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## G = C24.6Q8order 192 = 26·3

### 6th non-split extension by C24 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24.6Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C12.C8 — C24.6Q8
 Lower central C3 — C6 — C12 — C24 — C24.6Q8
 Upper central C1 — C2 — C2×C4 — C2×C8 — C8.C4

Generators and relations for C24.6Q8
G = < a,b,c | a24=1, b4=a12, c2=a21b2, bab-1=a7, cac-1=a5, cbc-1=a15b3 >

Character table of C24.6Q8

 class 1 2A 2B 3 4A 4B 4C 4D 6A 6B 8A 8B 8C 8D 8E 12A 12B 12C 16A 16B 16C 16D 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 2 2 2 2 24 24 2 4 2 2 4 8 8 2 2 4 12 12 12 12 4 4 4 4 8 8 8 8 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 i -i 1 -1 1 1 -1 -i i -1 -1 1 1 -1 -1 1 1 -1 1 -1 i i -i -i linear of order 4 ρ6 1 1 -1 1 -1 1 i -i 1 -1 1 1 -1 i -i -1 -1 1 -1 1 1 -1 1 -1 1 -1 -i -i i i linear of order 4 ρ7 1 1 -1 1 -1 1 -i i 1 -1 1 1 -1 -i i -1 -1 1 -1 1 1 -1 1 -1 1 -1 i i -i -i linear of order 4 ρ8 1 1 -1 1 -1 1 -i i 1 -1 1 1 -1 i -i -1 -1 1 1 -1 -1 1 1 -1 1 -1 -i -i i i linear of order 4 ρ9 2 2 2 2 2 2 0 0 2 2 -2 -2 -2 0 0 2 2 2 0 0 0 0 -2 -2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -1 2 2 0 0 -1 -1 2 2 2 2 2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 -1 2 2 0 0 -1 -1 2 2 2 -2 -2 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 -2 2 -2 2 0 0 2 -2 -2 -2 2 0 0 -2 -2 2 0 0 0 0 -2 2 -2 2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 2 -2 -1 -2 2 0 0 -1 1 -2 -2 2 0 0 1 1 -1 0 0 0 0 1 -1 1 -1 -√3 √3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ14 2 2 -2 -1 -2 2 0 0 -1 1 -2 -2 2 0 0 1 1 -1 0 0 0 0 1 -1 1 -1 √3 -√3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ15 2 2 -2 -1 -2 2 0 0 -1 1 2 2 -2 -2i 2i 1 1 -1 0 0 0 0 -1 1 -1 1 -i -i i i complex lifted from C4×S3 ρ16 2 2 -2 -1 -2 2 0 0 -1 1 2 2 -2 2i -2i 1 1 -1 0 0 0 0 -1 1 -1 1 i i -i -i complex lifted from C4×S3 ρ17 2 2 2 -1 2 2 0 0 -1 -1 -2 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 √-3 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ18 2 2 2 -1 2 2 0 0 -1 -1 -2 -2 -2 0 0 -1 -1 -1 0 0 0 0 1 1 1 1 -√-3 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ19 2 2 -2 2 2 -2 0 0 2 -2 0 0 0 0 0 2 2 -2 √-2 -√-2 √-2 -√-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ20 2 2 -2 2 2 -2 0 0 2 -2 0 0 0 0 0 2 2 -2 -√-2 √-2 -√-2 √-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ21 2 2 2 2 -2 -2 0 0 2 2 0 0 0 0 0 -2 -2 -2 -√-2 -√-2 √-2 √-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ22 2 2 2 2 -2 -2 0 0 2 2 0 0 0 0 0 -2 -2 -2 √-2 √-2 -√-2 -√-2 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ23 4 4 4 -2 -4 -4 0 0 -2 -2 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ24 4 4 -4 -2 4 -4 0 0 -2 2 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from D4.S3, Schur index 2 ρ25 4 -4 0 4 0 0 0 0 -4 0 2√-2 -2√-2 0 0 0 0 0 0 0 0 0 0 -2√-2 0 2√-2 0 0 0 0 0 complex lifted from C8.Q8 ρ26 4 -4 0 4 0 0 0 0 -4 0 -2√-2 2√-2 0 0 0 0 0 0 0 0 0 0 2√-2 0 -2√-2 0 0 0 0 0 complex lifted from C8.Q8 ρ27 4 -4 0 -2 0 0 0 0 2 0 -2√-2 2√-2 0 0 0 -2√3 2√3 0 0 0 0 0 -√-2 √-6 √-2 -√-6 0 0 0 0 complex faithful ρ28 4 -4 0 -2 0 0 0 0 2 0 2√-2 -2√-2 0 0 0 2√3 -2√3 0 0 0 0 0 √-2 √-6 -√-2 -√-6 0 0 0 0 complex faithful ρ29 4 -4 0 -2 0 0 0 0 2 0 -2√-2 2√-2 0 0 0 2√3 -2√3 0 0 0 0 0 -√-2 -√-6 √-2 √-6 0 0 0 0 complex faithful ρ30 4 -4 0 -2 0 0 0 0 2 0 2√-2 -2√-2 0 0 0 -2√3 2√3 0 0 0 0 0 √-2 -√-6 -√-2 √-6 0 0 0 0 complex faithful

Smallest permutation representation of C24.6Q8
On 48 points
Generators in S48
```(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)
(1 40 19 46 13 28 7 34)(2 47 20 29 14 35 8 41)(3 30 21 36 15 42 9 48)(4 37 22 43 16 25 10 31)(5 44 23 26 17 32 11 38)(6 27 24 33 18 39 12 45)
(1 34 16 37 7 40 22 43 13 46 4 25 19 28 10 31)(2 39 17 42 8 45 23 48 14 27 5 30 20 33 11 36)(3 44 18 47 9 26 24 29 15 32 6 35 21 38 12 41)```

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

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

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

Matrix representation of C24.6Q8 in GL6(𝔽97)

 0 1 0 0 0 0 96 1 0 0 0 0 0 0 57 57 0 0 0 0 40 57 0 0 0 0 0 0 40 57 0 0 0 0 40 40
,
 22 0 0 0 0 0 0 22 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 96 0 0 0 0 1 0 0 0
,
 0 96 0 0 0 0 96 0 0 0 0 0 0 0 0 0 40 40 0 0 0 0 40 57 0 0 1 0 0 0 0 0 0 96 0 0

`G:=sub<GL(6,GF(97))| [0,96,0,0,0,0,1,1,0,0,0,0,0,0,57,40,0,0,0,0,57,57,0,0,0,0,0,0,40,40,0,0,0,0,57,40],[22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,0,0,1,0,0,0,0,96,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,96,0,0,0,0,96,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,40,40,0,0,0,0,40,57,0,0] >;`

C24.6Q8 in GAP, Magma, Sage, TeX

`C_{24}._6Q_8`
`% in TeX`

`G:=Group("C24.6Q8");`
`// GroupNames label`

`G:=SmallGroup(192,53);`
`// by ID`

`G=gap.SmallGroup(192,53);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,365,36,758,184,346,80,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^4=a^12,c^2=a^21*b^2,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^15*b^3>;`
`// generators/relations`

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