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## G = C24.97D4order 192 = 26·3

### 20th non-split extension by C24 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C24.97D4
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C8×Dic3 — C24.97D4
 Lower central C3 — C6 — C12 — C24.97D4
 Upper central C1 — C8 — C2×C8 — M5(2)

Generators and relations for C24.97D4
G = < a,b,c | a24=1, b4=a12, c2=a9, bab-1=cac-1=a17, cbc-1=a12b3 >

Smallest permutation representation of C24.97D4
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 4 7 10 13 16 19 22)(2 21 8 3 14 9 20 15)(5 24 11 6 17 12 23 18)(25 46 43 40 37 34 31 28)(26 39 44 33 38 27 32 45)(29 42 47 36 41 30 35 48)
(1 46 10 31 19 40 4 25 13 34 22 43 7 28 16 37)(2 39 11 48 20 33 5 42 14 27 23 36 8 45 17 30)(3 32 12 41 21 26 6 35 15 44 24 29 9 38 18 47)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 4D 4E 4F 4G 6A 6B 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J 12A 12B 12C 16A 16B 16C 16D 16E 16F 16G 16H 24A 24B 24C 24D 24E 24F 48A ··· 48H order 1 2 2 3 4 4 4 4 4 4 4 6 6 8 8 8 8 8 8 8 8 8 8 12 12 12 16 16 16 16 16 16 16 16 24 24 24 24 24 24 48 ··· 48 size 1 1 2 2 1 1 2 6 6 6 6 2 4 1 1 1 1 2 2 6 6 6 6 2 2 4 4 4 4 4 12 12 12 12 2 2 2 2 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + - + - image C1 C2 C2 C2 C4 C4 C8 S3 D4 Q8 D6 M4(2) Dic6 C3⋊D4 C4×S3 S3×C8 C8⋊S3 C8.C8 C24.97D4 kernel C24.97D4 C12.C8 C8×Dic3 C3×M5(2) C2×C3⋊C8 C4×Dic3 C3⋊C8 M5(2) C24 C24 C2×C8 C2×C6 C8 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 8 1 1 1 1 2 2 2 2 4 4 8 4

Matrix representation of C24.97D4 in GL4(𝔽97) generated by

 0 96 0 0 1 96 0 0 0 0 33 0 0 0 0 33
,
 0 22 0 0 22 0 0 0 0 0 47 0 0 0 40 64
,
 39 29 0 0 68 58 0 0 0 0 64 66 0 0 81 33
`G:=sub<GL(4,GF(97))| [0,1,0,0,96,96,0,0,0,0,33,0,0,0,0,33],[0,22,0,0,22,0,0,0,0,0,47,40,0,0,0,64],[39,68,0,0,29,58,0,0,0,0,64,81,0,0,66,33] >;`

C24.97D4 in GAP, Magma, Sage, TeX

`C_{24}._{97}D_4`
`% in TeX`

`G:=Group("C24.97D4");`
`// GroupNames label`

`G:=SmallGroup(192,70);`
`// by ID`

`G=gap.SmallGroup(192,70);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,141,36,100,570,136,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^4=a^12,c^2=a^9,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^12*b^3>;`
`// generators/relations`

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