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## G = C8.25D12order 192 = 26·3

### 11st non-split extension by C8 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C8.25D12
 Chief series C1 — C3 — C6 — C12 — C24 — C2×C24 — C2×C8⋊S3 — C8.25D12
 Lower central C3 — C6 — C2×C6 — C8.25D12
 Upper central C1 — C4 — C2×C8 — M5(2)

Generators and relations for C8.25D12
G = < a,b,c | a8=1, b12=a6, c2=a, bab-1=a5, ac=ca, cbc-1=a3b11 >

Subgroups: 152 in 58 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, C12, D6, C2×C6, C16, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, M5(2), M5(2), C2×M4(2), C3⋊C16, C48, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, C23.C8, C12.C8, C3×M5(2), C2×C8⋊S3, C8.25D12
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), C4×S3, D12, C3⋊D4, C22⋊C8, S3×C8, C8⋊S3, D6⋊C4, C23.C8, D6⋊C8, C8.25D12

Smallest permutation representation of C8.25D12
On 48 points
Generators in S48
```(1 19 37 7 25 43 13 31)(2 44 38 32 26 20 14 8)(3 21 39 9 27 45 15 33)(4 46 40 34 28 22 16 10)(5 23 41 11 29 47 17 35)(6 48 42 36 30 24 18 12)
(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 42 19 36 37 30 7 24 25 18 43 12 13 6 31 48)(2 11 44 29 38 47 32 17 26 35 20 5 14 23 8 41)(3 4 21 46 39 40 9 34 27 28 45 22 15 16 33 10)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C8 C8 S3 D4 D6 M4(2) D12 C3⋊D4 C4×S3 C8⋊S3 S3×C8 C23.C8 C8.25D12 kernel C8.25D12 C12.C8 C3×M5(2) C2×C8⋊S3 C2×C3⋊C8 S3×C2×C4 C2×Dic3 C22×S3 M5(2) C24 C2×C8 C12 C8 C8 C2×C4 C4 C22 C3 C1 # reps 1 1 1 1 2 2 4 4 1 2 1 2 2 2 2 4 4 2 4

Matrix representation of C8.25D12 in GL4(𝔽5) generated by

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

C8.25D12 in GAP, Magma, Sage, TeX

`C_8._{25}D_{12}`
`% in TeX`

`G:=Group("C8.25D12");`
`// GroupNames label`

`G:=SmallGroup(192,73);`
`// by ID`

`G=gap.SmallGroup(192,73);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,758,100,570,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^8=1,b^12=a^6,c^2=a,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^3*b^11>;`
`// generators/relations`

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