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## G = C12.20C42order 192 = 26·3

### 13rd non-split extension by C12 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.20C42
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C23.26D6 — C12.20C42
 Lower central C3 — C12 — C12.20C42
 Upper central C1 — C4 — C2×M4(2)

Generators and relations for C12.20C42
G = < a,b,c | a12=b4=1, c4=a6, bab-1=a5, ac=ca, cbc-1=a9b >

Subgroups: 232 in 94 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C42⋊C2, C2×M4(2), C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C22×C12, C4.9C42, C23.26D6, C6×M4(2), C12.20C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4.9C42, C6.C42, C12.20C42

Smallest permutation representation of C12.20C42
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 42 13 26)(2 47 14 31)(3 40 15 36)(4 45 16 29)(5 38 17 34)(6 43 18 27)(7 48 19 32)(8 41 20 25)(9 46 21 30)(10 39 22 35)(11 44 23 28)(12 37 24 33)
(1 48 16 29 7 42 22 35)(2 37 17 30 8 43 23 36)(3 38 18 31 9 44 24 25)(4 39 19 32 10 45 13 26)(5 40 20 33 11 46 14 27)(6 41 21 34 12 47 15 28)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F ··· 4M 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 3 4 4 4 4 4 4 ··· 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 2 2 2 2 1 1 2 2 2 12 ··· 12 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + - + - + image C1 C2 C2 C4 C4 S3 D4 Q8 D4 Dic3 D6 Dic6 C4×S3 C3⋊D4 D12 C4.9C42 C12.20C42 kernel C12.20C42 C23.26D6 C6×M4(2) C4×Dic3 C2×C24 C2×M4(2) C2×C12 C2×C12 C22×C6 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C3 C1 # reps 1 2 1 8 4 1 2 1 1 2 1 2 4 4 2 2 4

Matrix representation of C12.20C42 in GL4(𝔽73) generated by

 0 27 0 0 46 46 0 0 0 0 0 27 0 0 46 46
,
 0 0 0 46 0 0 46 0 14 7 0 0 66 59 0 0
,
 0 0 1 0 0 0 0 1 7 14 0 0 59 66 0 0
`G:=sub<GL(4,GF(73))| [0,46,0,0,27,46,0,0,0,0,0,46,0,0,27,46],[0,0,14,66,0,0,7,59,0,46,0,0,46,0,0,0],[0,0,7,59,0,0,14,66,1,0,0,0,0,1,0,0] >;`

C12.20C42 in GAP, Magma, Sage, TeX

`C_{12}._{20}C_4^2`
`% in TeX`

`G:=Group("C12.20C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,116);`
`// by ID`

`G=gap.SmallGroup(192,116);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1123,136,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^4=1,c^4=a^6,b*a*b^-1=a^5,a*c=c*a,c*b*c^-1=a^9*b>;`
`// generators/relations`

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