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## G = C42.7D6order 192 = 26·3

### 7th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.7D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4×C12 — C42.S3 — C42.7D6
 Lower central C3 — C2×C6 — C2×C12 — C42.7D6
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.7D6
G = < a,b,c,d | a4=b4=c6=1, d2=ab-1, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, dbd-1=a2b-1, dcd-1=a2bc-1 >

Subgroups: 176 in 70 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, C2×D4, C2×Q8, C3⋊C8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C8⋊C4, C4.4D4, C2×C3⋊C8, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C42.C22, C42.S3, C3×C4.4D4, C42.7D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C2×Dic3, C3⋊D4, C4.D4, C4≀C2, C6.D4, C42.C22, C12.D4, Q83Dic3, C42.7D6

Smallest permutation representation of C42.7D6
On 96 points
Generators in S96
```(1 57 32 47)(2 62 25 44)(3 59 26 41)(4 64 27 46)(5 61 28 43)(6 58 29 48)(7 63 30 45)(8 60 31 42)(9 80 35 91)(10 77 36 96)(11 74 37 93)(12 79 38 90)(13 76 39 95)(14 73 40 92)(15 78 33 89)(16 75 34 94)(17 72 86 50)(18 69 87 55)(19 66 88 52)(20 71 81 49)(21 68 82 54)(22 65 83 51)(23 70 84 56)(24 67 85 53)
(1 63 28 41)(2 60 29 46)(3 57 30 43)(4 62 31 48)(5 59 32 45)(6 64 25 42)(7 61 26 47)(8 58 27 44)(9 78 39 93)(10 75 40 90)(11 80 33 95)(12 77 34 92)(13 74 35 89)(14 79 36 94)(15 76 37 91)(16 73 38 96)(17 70 82 52)(18 67 83 49)(19 72 84 54)(20 69 85 51)(21 66 86 56)(22 71 87 53)(23 68 88 50)(24 65 81 55)
(1 24 9)(2 75 17 60 10 70)(3 87 11 26 18 37)(4 96 19 44 12 50)(5 20 13)(6 79 21 64 14 66)(7 83 15 30 22 33)(8 92 23 48 16 54)(25 94 86 42 36 56)(27 77 88 62 38 72)(28 81 39)(29 90 82 46 40 52)(31 73 84 58 34 68)(32 85 35)(41 65 93 63 55 78)(43 53 95 47 49 91)(45 69 89 59 51 74)(57 71 80 61 67 76)
(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)```

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + - - + image C1 C2 C2 C4 C4 S3 D4 D6 Dic3 Dic3 C3⋊D4 C4≀C2 C4.D4 C12.D4 Q8⋊3Dic3 kernel C42.7D6 C42.S3 C3×C4.4D4 C6×D4 C6×Q8 C4.4D4 C2×C12 C42 C2×D4 C2×Q8 C2×C4 C6 C6 C2 C2 # reps 1 2 1 2 2 1 2 1 1 1 4 8 1 2 4

Matrix representation of C42.7D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 46 0 0 0 0 27 0 0 0 0 0 0 0 46 0 0 0 0 0 0 46
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 64 0 0 0 0 0 0 65 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72
,
 0 8 0 0 0 0 9 0 0 0 0 0 0 0 13 13 0 0 0 0 13 60 0 0 0 0 0 0 60 13 0 0 0 0 60 60

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,46,0,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[64,0,0,0,0,0,0,65,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[0,9,0,0,0,0,8,0,0,0,0,0,0,0,13,13,0,0,0,0,13,60,0,0,0,0,0,0,60,60,0,0,0,0,13,60] >;`

C42.7D6 in GAP, Magma, Sage, TeX

`C_4^2._7D_6`
`% in TeX`

`G:=Group("C4^2.7D6");`
`// GroupNames label`

`G:=SmallGroup(192,99);`
`// by ID`

`G=gap.SmallGroup(192,99);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,268,1571,570,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a*b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=a^2*b*c^-1>;`
`// generators/relations`

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