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G = C12.4C42order 192 = 26·3

4th non-split extension by C12 of C42 acting via C42/C22=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.4C42
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C22×C12 — C22×C3⋊C8 — C12.4C42
 Lower central C3 — C6 — C12 — C12.4C42
 Upper central C1 — C2×C4 — C22×C4 — C2×M4(2)

Generators and relations for C12.4C42
G = < a,b,c | a12=1, b4=c4=a6, bab-1=a5, cac-1=a7, cbc-1=a3b >

Subgroups: 168 in 90 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, C23, C12, C2×C6, C2×C6, C2×C8, M4(2), M4(2), C22×C4, C3⋊C8, C24, C2×C12, C22×C6, C22×C8, C2×M4(2), C2×M4(2), C2×C3⋊C8, C2×C3⋊C8, C4.Dic3, C4.Dic3, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C4.C42, C22×C3⋊C8, C2×C4.Dic3, C6×M4(2), C12.4C42
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, C8.C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4.C42, C12.53D4, C6.C42, C12.4C42

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E ··· 8L 8M 8N 8O 8P 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 6 6 6 6 6 8 8 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 2 2 2 4 4 4 4 4 4 6 ··· 6 12 12 12 12 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 4 type + + + + + + - - + + - image C1 C2 C2 C2 C4 C4 C4 S3 D4 Q8 Dic3 D6 C4×S3 D12 C3⋊D4 Dic6 C8.C4 C12.53D4 kernel C12.4C42 C22×C3⋊C8 C2×C4.Dic3 C6×M4(2) C2×C3⋊C8 C4.Dic3 C3×M4(2) C2×M4(2) 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 3 1 2 1 4 2 4 2 8 4

Matrix representation of C12.4C42 in GL4(𝔽73) generated by

 0 1 0 0 72 72 0 0 0 0 46 0 0 0 1 27
,
 71 53 0 0 55 2 0 0 0 0 10 0 0 0 67 51
,
 46 0 0 0 0 46 0 0 0 0 27 71 0 0 13 46
`G:=sub<GL(4,GF(73))| [0,72,0,0,1,72,0,0,0,0,46,1,0,0,0,27],[71,55,0,0,53,2,0,0,0,0,10,67,0,0,0,51],[46,0,0,0,0,46,0,0,0,0,27,13,0,0,71,46] >;`

C12.4C42 in GAP, Magma, Sage, TeX

`C_{12}._4C_4^2`
`% in TeX`

`G:=Group("C12.4C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,117);`
`// by ID`

`G=gap.SmallGroup(192,117);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,365,36,184,570,136,6278]);`
`// 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,c*b*c^-1=a^3*b>;`
`// generators/relations`

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