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## G = C42.61D6order 192 = 26·3

### 61st non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.61D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C4×Dic6 — C42.61D6
 Lower central C3 — C6 — C2×C12 — C42.61D6
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.61D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 288 in 112 conjugacy classes, 41 normal (39 characteristic)
C1, C2 [×3], C2, C3, C4 [×2], C4 [×6], C22, C22 [×3], C6 [×3], C6, C8 [×2], C2×C4 [×3], C2×C4 [×3], D4 [×2], Q8 [×5], C23, Dic3 [×3], C12 [×2], C12 [×3], C2×C6, C2×C6 [×3], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], SD16 [×2], Q16 [×2], C2×D4, C2×Q8, C2×Q8, C3⋊C8 [×2], Dic6 [×2], Dic6, C2×Dic3 [×2], C2×C12 [×3], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C2×C3⋊C8 [×2], C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3 [×2], C3⋊Q16 [×2], C4×C12, C3×C22⋊C4 [×2], C2×Dic6, C6×D4, C6×Q8, Q8.D4, C12⋊C8, D4⋊Dic3, Q82Dic3, C4×Dic6, C2×D4.S3, C2×C3⋊Q16, C3×C4.4D4, C42.61D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3, C4⋊D4, C4○D8, C8.C22, S3×D4, D42S3, C2×C3⋊D4, Q8.D4, D63D4, Q8.13D6, Q8.14D6, C42.61D6

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 C4○D8 C8.C22 S3×D4 D4⋊2S3 Q8.13D6 Q8.14D6 kernel C42.61D6 C12⋊C8 D4⋊Dic3 Q8⋊2Dic3 C4×Dic6 C2×D4.S3 C2×C3⋊Q16 C3×C4.4D4 C4.4D4 Dic6 C2×C12 C42 C2×D4 C2×Q8 C12 C2×C4 C6 C6 C4 C4 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 4 4 1 1 1 2 2

Matrix representation of C42.61D6 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 0 0 0 0 0 0 46 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 71 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72
,
 11 71 0 0 0 0 60 62 0 0 0 0 0 0 0 32 0 0 0 0 57 32 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[11,60,0,0,0,0,71,62,0,0,0,0,0,0,0,57,0,0,0,0,32,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C42.61D6 in GAP, Magma, Sage, TeX

`C_4^2._{61}D_6`
`% in TeX`

`G:=Group("C4^2.61D6");`
`// GroupNames label`

`G:=SmallGroup(192,613);`
`// by ID`

`G=gap.SmallGroup(192,613);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,254,219,1123,297,136,6278]);`
`// Polycyclic`

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

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