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## G = C12.3C42order 192 = 26·3

### 3rd non-split extension by C12 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.3C42
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C22×C12 — C23.26D6 — C12.3C42
 Lower central C3 — C6 — C12 — C12.3C42
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C12.3C42
G = < a,b,c | a12=b4=1, c4=a6, bab-1=a-1, cac-1=a7, cbc-1=a9b >

Subgroups: 264 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C22×C6, C2×C42, C42⋊C2, C2×M4(2), C4×Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C3×M4(2), C22×Dic3, C22×C12, C426C4, C2×C4×Dic3, C23.26D6, C6×M4(2), C12.3C42
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≀C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C426C4, D12⋊C4, C6.C42, C12.3C42

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 4Q 4R 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 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 1 1 2 2 2 1 1 1 1 2 2 6 ··· 6 12 12 12 12 2 2 2 4 4 4 4 4 4 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 C4 S3 D4 Q8 D4 Dic3 D6 Dic6 C4×S3 C3⋊D4 D12 C4≀C2 D12⋊C4 kernel C12.3C42 C2×C4×Dic3 C23.26D6 C6×M4(2) C4×Dic3 C4⋊Dic3 C3×M4(2) C2×M4(2) C2×C12 C2×C12 C22×C6 M4(2) C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 1 1 1 4 4 4 1 2 1 1 2 1 2 4 4 2 8 4

Matrix representation of C12.3C42 in GL4(𝔽73) generated by

 0 72 0 0 1 1 0 0 0 0 46 0 0 0 1 27
,
 30 60 0 0 30 43 0 0 0 0 72 19 0 0 46 1
,
 66 59 0 0 14 7 0 0 0 0 27 71 0 0 13 46
`G:=sub<GL(4,GF(73))| [0,1,0,0,72,1,0,0,0,0,46,1,0,0,0,27],[30,30,0,0,60,43,0,0,0,0,72,46,0,0,19,1],[66,14,0,0,59,7,0,0,0,0,27,13,0,0,71,46] >;`

C12.3C42 in GAP, Magma, Sage, TeX

`C_{12}._3C_4^2`
`% in TeX`

`G:=Group("C12.3C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,114);`
`// by ID`

`G=gap.SmallGroup(192,114);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,136,1684,851,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=b^4=1,c^4=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;`
`// generators/relations`

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