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## G = C12.12C42order 192 = 26·3

### 5th non-split extension by C12 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C12.12C42
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×Dic3 — C23.26D6 — C12.12C42
 Lower central C3 — C6 — C12.12C42
 Upper central C1 — C2×C8 — C22×C8

Generators and relations for C12.12C42
G = < a,b,c | a12=b4=1, c4=a6, bab-1=a-1, ac=ca, bc=cb >

Subgroups: 216 in 130 conjugacy classes, 87 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C4×C8, C8⋊C4, C42⋊C2, C22×C8, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C2×C24, C22×C12, C82M4(2), C8×Dic3, C24⋊C4, C2×C4.Dic3, C23.26D6, C22×C24, C12.12C42
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C42, C22×C4, C4×S3, C2×Dic3, C22×S3, C2×C42, C8○D4, C4×Dic3, S3×C2×C4, C22×Dic3, C82M4(2), C8○D12, C2×C4×Dic3, C12.12C42

Smallest permutation representation of C12.12C42
On 96 points
Generators in S96
```(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 67 16 55)(2 66 17 54)(3 65 18 53)(4 64 19 52)(5 63 20 51)(6 62 21 50)(7 61 22 49)(8 72 23 60)(9 71 24 59)(10 70 13 58)(11 69 14 57)(12 68 15 56)(25 94 46 76)(26 93 47 75)(27 92 48 74)(28 91 37 73)(29 90 38 84)(30 89 39 83)(31 88 40 82)(32 87 41 81)(33 86 42 80)(34 85 43 79)(35 96 44 78)(36 95 45 77)
(1 25 10 34 7 31 4 28)(2 26 11 35 8 32 5 29)(3 27 12 36 9 33 6 30)(13 43 22 40 19 37 16 46)(14 44 23 41 20 38 17 47)(15 45 24 42 21 39 18 48)(49 82 52 73 55 76 58 79)(50 83 53 74 56 77 59 80)(51 84 54 75 57 78 60 81)(61 88 64 91 67 94 70 85)(62 89 65 92 68 95 71 86)(63 90 66 93 69 96 72 87)```

`G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,67,16,55)(2,66,17,54)(3,65,18,53)(4,64,19,52)(5,63,20,51)(6,62,21,50)(7,61,22,49)(8,72,23,60)(9,71,24,59)(10,70,13,58)(11,69,14,57)(12,68,15,56)(25,94,46,76)(26,93,47,75)(27,92,48,74)(28,91,37,73)(29,90,38,84)(30,89,39,83)(31,88,40,82)(32,87,41,81)(33,86,42,80)(34,85,43,79)(35,96,44,78)(36,95,45,77), (1,25,10,34,7,31,4,28)(2,26,11,35,8,32,5,29)(3,27,12,36,9,33,6,30)(13,43,22,40,19,37,16,46)(14,44,23,41,20,38,17,47)(15,45,24,42,21,39,18,48)(49,82,52,73,55,76,58,79)(50,83,53,74,56,77,59,80)(51,84,54,75,57,78,60,81)(61,88,64,91,67,94,70,85)(62,89,65,92,68,95,71,86)(63,90,66,93,69,96,72,87)>;`

`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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,67,16,55)(2,66,17,54)(3,65,18,53)(4,64,19,52)(5,63,20,51)(6,62,21,50)(7,61,22,49)(8,72,23,60)(9,71,24,59)(10,70,13,58)(11,69,14,57)(12,68,15,56)(25,94,46,76)(26,93,47,75)(27,92,48,74)(28,91,37,73)(29,90,38,84)(30,89,39,83)(31,88,40,82)(32,87,41,81)(33,86,42,80)(34,85,43,79)(35,96,44,78)(36,95,45,77), (1,25,10,34,7,31,4,28)(2,26,11,35,8,32,5,29)(3,27,12,36,9,33,6,30)(13,43,22,40,19,37,16,46)(14,44,23,41,20,38,17,47)(15,45,24,42,21,39,18,48)(49,82,52,73,55,76,58,79)(50,83,53,74,56,77,59,80)(51,84,54,75,57,78,60,81)(61,88,64,91,67,94,70,85)(62,89,65,92,68,95,71,86)(63,90,66,93,69,96,72,87) );`

`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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,67,16,55),(2,66,17,54),(3,65,18,53),(4,64,19,52),(5,63,20,51),(6,62,21,50),(7,61,22,49),(8,72,23,60),(9,71,24,59),(10,70,13,58),(11,69,14,57),(12,68,15,56),(25,94,46,76),(26,93,47,75),(27,92,48,74),(28,91,37,73),(29,90,38,84),(30,89,39,83),(31,88,40,82),(32,87,41,81),(33,86,42,80),(34,85,43,79),(35,96,44,78),(36,95,45,77)], [(1,25,10,34,7,31,4,28),(2,26,11,35,8,32,5,29),(3,27,12,36,9,33,6,30),(13,43,22,40,19,37,16,46),(14,44,23,41,20,38,17,47),(15,45,24,42,21,39,18,48),(49,82,52,73,55,76,58,79),(50,83,53,74,56,77,59,80),(51,84,54,75,57,78,60,81),(61,88,64,91,67,94,70,85),(62,89,65,92,68,95,71,86),(63,90,66,93,69,96,72,87)]])`

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 6A ··· 6G 8A ··· 8H 8I 8J 8K 8L 8M ··· 8T 12A ··· 12H 24A ··· 24P order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 6 ··· 6 8 ··· 8 8 8 8 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 2 1 1 1 1 2 2 6 ··· 6 2 ··· 2 1 ··· 1 2 2 2 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 Dic3 D6 D6 C4×S3 C4×S3 C8○D4 C8○D12 kernel C12.12C42 C8×Dic3 C24⋊C4 C2×C4.Dic3 C23.26D6 C22×C24 C4.Dic3 C4⋊Dic3 C6.D4 C2×C24 C22×C8 C2×C8 C2×C8 C22×C4 C2×C4 C23 C6 C2 # reps 1 2 2 1 1 1 8 4 4 8 1 4 2 1 6 2 8 16

Matrix representation of C12.12C42 in GL4(𝔽73) generated by

 27 0 0 0 0 46 0 0 0 0 49 0 0 0 0 3
,
 0 1 0 0 72 0 0 0 0 0 0 1 0 0 1 0
,
 63 0 0 0 0 63 0 0 0 0 10 0 0 0 0 10
`G:=sub<GL(4,GF(73))| [27,0,0,0,0,46,0,0,0,0,49,0,0,0,0,3],[0,72,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[63,0,0,0,0,63,0,0,0,0,10,0,0,0,0,10] >;`

C12.12C42 in GAP, Magma, Sage, TeX

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

`G:=Group("C12.12C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,660);`
`// by ID`

`G=gap.SmallGroup(192,660);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,100,102,6278]);`
`// Polycyclic`

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

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