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## G = C12.7C42order 192 = 26·3

### 7th non-split extension by C12 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C12.7C42
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C22×C3⋊C8 — C12.7C42
 Lower central C3 — C6 — C12.7C42
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C12.7C42
G = < a,b,c | a12=1, b4=c4=a6, bab-1=a5, cac-1=a7, bc=cb >

Subgroups: 216 in 130 conjugacy classes, 87 normal (25 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, 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, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C22×C12, C82M4(2), C8×Dic3, C24⋊C4, C22×C3⋊C8, C23.26D6, C6×M4(2), C12.7C42
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), D12.C4, C2×C4×Dic3, C12.7C42

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

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 6A 6B 6C 6D 6E 8A ··· 8H 8I ··· 8P 8Q 8R 8S 8T 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 6 6 6 6 6 8 ··· 8 8 ··· 8 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 2 2 2 4 4 2 ··· 2 3 ··· 3 6 6 6 6 2 2 2 2 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D6 Dic3 D6 C4×S3 C4×S3 C8○D4 D12.C4 kernel C12.7C42 C8×Dic3 C24⋊C4 C22×C3⋊C8 C23.26D6 C6×M4(2) C2×C3⋊C8 C4⋊Dic3 C6.D4 C3×M4(2) C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C6 C2 # reps 1 2 2 1 1 1 8 4 4 8 1 2 4 1 6 2 8 4

Matrix representation of C12.7C42 in GL4(𝔽73) generated by

 0 1 0 0 72 72 0 0 0 0 27 0 0 0 26 46
,
 31 3 0 0 45 42 0 0 0 0 63 0 0 0 0 63
,
 46 0 0 0 0 46 0 0 0 0 28 71 0 0 50 45
`G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,27,26,0,0,0,46],[31,45,0,0,3,42,0,0,0,0,63,0,0,0,0,63],[46,0,0,0,0,46,0,0,0,0,28,50,0,0,71,45] >;`

C12.7C42 in GAP, Magma, Sage, TeX

`C_{12}._7C_4^2`
`% in TeX`

`G:=Group("C12.7C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,681);`
`// by ID`

`G=gap.SmallGroup(192,681);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,387,100,136,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^4=c^4=a^6,b*a*b^-1=a^5,c*a*c^-1=a^7,b*c=c*b>;`
`// generators/relations`

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