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## G = C42.171D6order 192 = 26·3

### 171st non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C42.171D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×S3×Q8 — C42.171D6
 Lower central C3 — C2×C6 — C42.171D6
 Upper central C1 — C22 — C4⋊Q8

Generators and relations for C42.171D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >

Subgroups: 688 in 270 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×12], C22, C22 [×10], S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×6], C2×C4 [×17], D4 [×6], Q8 [×10], C23 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×6], D6 [×2], D6 [×8], C2×C6, C42, C42, C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4 [×5], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×7], C4○D4 [×4], Dic6 [×6], C4×S3 [×4], C4×S3 [×8], D12 [×6], C2×Dic3, C2×Dic3 [×4], C2×C12, C2×C12 [×6], C3×Q8 [×4], C22×S3, C22×S3 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8, C4⋊Q8, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×6], D6⋊C4 [×10], C4×C12, C3×C4⋊C4 [×4], C2×Dic6, C2×Dic6 [×2], S3×C2×C4, S3×C2×C4 [×4], C2×D12, C2×D12 [×2], S3×Q8 [×4], Q83S3 [×4], C6×Q8 [×2], C23.38C23, C422S3, C427S3, D6.D4 [×4], D6⋊Q8 [×4], Dic3⋊Q8, C12.23D4, C3×C4⋊Q8, C2×S3×Q8, C2×Q83S3, C42.171D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2- 1+4 [×2], S3×D4 [×2], S3×C23, C23.38C23, C2×S3×D4, Q8.15D6 [×2], C42.171D6

Smallest permutation representation of C42.171D6
On 96 points
Generators in S96
```(1 70 7 64)(2 65 8 71)(3 72 9 66)(4 67 10 61)(5 62 11 68)(6 69 12 63)(13 59 19 53)(14 54 20 60)(15 49 21 55)(16 56 22 50)(17 51 23 57)(18 58 24 52)(25 39 31 45)(26 46 32 40)(27 41 33 47)(28 48 34 42)(29 43 35 37)(30 38 36 44)(73 87 79 93)(74 94 80 88)(75 89 81 95)(76 96 82 90)(77 91 83 85)(78 86 84 92)
(1 59 85 42)(2 43 86 60)(3 49 87 44)(4 45 88 50)(5 51 89 46)(6 47 90 52)(7 53 91 48)(8 37 92 54)(9 55 93 38)(10 39 94 56)(11 57 95 40)(12 41 96 58)(13 83 34 64)(14 65 35 84)(15 73 36 66)(16 67 25 74)(17 75 26 68)(18 69 27 76)(19 77 28 70)(20 71 29 78)(21 79 30 72)(22 61 31 80)(23 81 32 62)(24 63 33 82)
(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 38 7 44)(2 43 8 37)(3 48 9 42)(4 41 10 47)(5 46 11 40)(6 39 12 45)(13 79 19 73)(14 84 20 78)(15 77 21 83)(16 82 22 76)(17 75 23 81)(18 80 24 74)(25 63 31 69)(26 68 32 62)(27 61 33 67)(28 66 34 72)(29 71 35 65)(30 64 36 70)(49 85 55 91)(50 90 56 96)(51 95 57 89)(52 88 58 94)(53 93 59 87)(54 86 60 92)```

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

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

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

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 2- 1+4 S3×D4 Q8.15D6 kernel C42.171D6 C42⋊2S3 C42⋊7S3 D6.D4 D6⋊Q8 Dic3⋊Q8 C12.23D4 C3×C4⋊Q8 C2×S3×Q8 C2×Q8⋊3S3 C4⋊Q8 C4×S3 C42 C4⋊C4 C2×Q8 C6 C4 C2 # reps 1 1 1 4 4 1 1 1 1 1 1 4 1 4 2 2 2 4

Matrix representation of C42.171D6 in GL8(𝔽13)

 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 3 4 8 6 0 0 0 0 9 3 6 5 0 0 0 0 4 0 10 4 0 0 0 0 0 9 9 10
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0
,
 0 12 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 11 0 0 0 0 0 12 0 0 11 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0
,
 12 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 12 0 0 0 0 0 1 1 0

`G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,3,9,4,0,0,0,0,0,4,3,0,9,0,0,0,0,8,6,10,9,0,0,0,0,6,5,4,10],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0],[0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,12,0,0,1,0,0,0,0,11,0,0,1,0,0,0,0,0,11,1,0],[12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0] >;`

C42.171D6 in GAP, Magma, Sage, TeX

`C_4^2._{171}D_6`
`% in TeX`

`G:=Group("C4^2.171D6");`
`// GroupNames label`

`G:=SmallGroup(192,1283);`
`// by ID`

`G=gap.SmallGroup(192,1283);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,100,675,297,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;`
`// generators/relations`

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