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## G = C12.10C42order 192 = 26·3

### 3rd non-split extension by C12 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C12.10C42
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C22×C12 — C2×C4.Dic3 — C12.10C42
 Lower central C3 — C6 — C12 — C12.10C42
 Upper central C1 — C2×C4 — C22×C4 — C22×C8

Generators and relations for C12.10C42
G = < a,b,c | a12=1, b4=c4=a6, bab-1=a-1, ac=ca, cbc-1=a3b >

Subgroups: 168 in 90 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C2×C12, C2×C12, C22×C6, C22×C8, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4.Dic3, C2×C24, C2×C24, C22×C12, C4.C42, C2×C4.Dic3, C22×C24, C12.10C42
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, C24.C4, C6.C42, C12.10C42

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + + - image C1 C2 C2 C4 C4 S3 D4 Q8 Dic3 D6 C4×S3 D12 C3⋊D4 Dic6 C8.C4 C24.C4 kernel C12.10C42 C2×C4.Dic3 C22×C24 C4.Dic3 C2×C24 C22×C8 C2×C12 C22×C6 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 2 1 8 4 1 3 1 2 1 4 2 4 2 8 16

Matrix representation of C12.10C42 in GL3(𝔽73) generated by

 1 0 0 0 24 0 0 0 70
,
 46 0 0 0 0 1 0 27 0
,
 46 0 0 0 63 0 0 0 51
`G:=sub<GL(3,GF(73))| [1,0,0,0,24,0,0,0,70],[46,0,0,0,0,27,0,1,0],[46,0,0,0,63,0,0,0,51] >;`

C12.10C42 in GAP, Magma, Sage, TeX

`C_{12}._{10}C_4^2`
`% in TeX`

`G:=Group("C12.10C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,111);`
`// by ID`

`G=gap.SmallGroup(192,111);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,176,184,1123,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^12=1,b^4=c^4=a^6,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;`
`// generators/relations`

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