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## G = C24.Q8order 192 = 26·3

### 1st non-split extension by C24 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24.Q8
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C24 — C8⋊Dic3 — C24.Q8
 Lower central C3 — C6 — C12 — C24 — C24.Q8
 Upper central C1 — C2 — C2×C4 — C2×C8 — M5(2)

Generators and relations for C24.Q8
G = < a,b,c | a24=1, b4=a18, c2=a15b2, bab-1=a13, cac-1=a11, cbc-1=a12b3 >

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

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

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

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

36 conjugacy classes

 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 48A ··· 48H order 1 2 2 3 4 4 4 4 6 6 8 8 8 8 8 12 12 12 16 16 16 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 2 2 2 24 24 2 4 2 2 4 24 24 2 2 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + - + - + image C1 C2 C2 C2 C4 S3 Q8 D4 Dic3 D6 SD16 SD16 Dic6 D12 C24⋊C2 C24⋊C2 C8.Q8 C24.Q8 kernel C24.Q8 C8⋊Dic3 C24.C4 C3×M5(2) C48 M5(2) C24 C2×C12 C16 C2×C8 C12 C2×C6 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 4 1 1 1 2 1 2 2 2 2 4 4 2 4

Matrix representation of C24.Q8 in GL4(𝔽97) generated by

 8 53 0 0 44 61 0 0 23 49 36 44 13 84 53 89
,
 0 0 96 1 13 3 95 96 36 18 26 68 72 26 26 68
,
 0 1 0 0 1 0 0 0 0 24 53 89 10 86 36 44
`G:=sub<GL(4,GF(97))| [8,44,23,13,53,61,49,84,0,0,36,53,0,0,44,89],[0,13,36,72,0,3,18,26,96,95,26,26,1,96,68,68],[0,1,0,10,1,0,24,86,0,0,53,36,0,0,89,44] >;`

C24.Q8 in GAP, Magma, Sage, TeX

`C_{24}.Q_8`
`% in TeX`

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

`G:=SmallGroup(192,72);`
`// by ID`

`G=gap.SmallGroup(192,72);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,64,387,675,80,1684,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^4=a^18,c^2=a^15*b^2,b*a*b^-1=a^13,c*a*c^-1=a^11,c*b*c^-1=a^12*b^3>;`
`// generators/relations`

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