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## G = C42.64D6order 192 = 26·3

### 64th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.64D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C4⋊D12 — C42.64D6
 Lower central C3 — C6 — C2×C12 — C42.64D6
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.64D6
G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1b2, dad-1=ab2, bd=db, dcd-1=b-1c-1 >

Subgroups: 528 in 144 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×2], C4 [×3], C22, C22 [×9], S3 [×2], C6, C6 [×2], C6, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], C12 [×2], C12 [×3], D6 [×6], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×4], SD16 [×4], C2×D4, C2×D4 [×4], C2×Q8, C3⋊C8 [×4], D12 [×8], C2×C12, C2×C12 [×2], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×S3 [×2], C22×C6, C8⋊C4, C4.4D4, C41D4, C2×D8 [×2], C2×SD16 [×2], C2×C3⋊C8 [×2], D4⋊S3 [×4], Q82S3 [×4], C4×C12, C3×C22⋊C4 [×2], C2×D12 [×2], C2×D12 [×2], C6×D4, C6×Q8, C83D4, C42.S3, C4⋊D12, C2×D4⋊S3 [×2], C2×Q82S3 [×2], C3×C4.4D4, C42.64D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C8⋊C22 [×2], S3×D4 [×2], C2×C3⋊D4, C83D4, C123D4, D4⋊D6 [×2], C42.64D6

Character table of C42.64D6

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 12G 12H size 1 1 1 1 8 24 24 2 2 2 4 4 8 2 2 2 8 8 12 12 12 12 4 4 4 4 4 4 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ6 1 1 1 1 -1 1 -1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 1 -1 1 1 1 linear of order 2 ρ7 1 1 1 1 -1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 -1 linear of order 2 ρ9 2 2 -2 -2 0 0 0 2 2 -2 0 0 0 -2 -2 2 0 0 0 2 -2 0 0 0 0 -2 0 2 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 0 0 -1 2 2 -2 -2 -2 -1 -1 -1 -1 -1 0 0 0 0 1 1 1 -1 1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 2 0 0 -1 2 2 2 2 2 -1 -1 -1 -1 -1 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 2 2 2 0 0 0 2 -2 -2 2 -2 0 2 2 2 0 0 0 0 0 0 2 -2 -2 -2 2 -2 0 0 orthogonal lifted from D4 ρ13 2 2 -2 -2 0 0 0 2 -2 2 0 0 0 -2 -2 2 0 0 2 0 0 -2 0 0 0 2 0 -2 0 0 orthogonal lifted from D4 ρ14 2 2 -2 -2 0 0 0 2 -2 2 0 0 0 -2 -2 2 0 0 -2 0 0 2 0 0 0 2 0 -2 0 0 orthogonal lifted from D4 ρ15 2 2 2 2 0 0 0 2 -2 -2 -2 2 0 2 2 2 0 0 0 0 0 0 -2 2 2 -2 -2 -2 0 0 orthogonal lifted from D4 ρ16 2 2 2 2 -2 0 0 -1 2 2 -2 -2 2 -1 -1 -1 1 1 0 0 0 0 1 1 1 -1 1 -1 -1 -1 orthogonal lifted from D6 ρ17 2 2 2 2 -2 0 0 -1 2 2 2 2 -2 -1 -1 -1 1 1 0 0 0 0 -1 -1 -1 -1 -1 -1 1 1 orthogonal lifted from D6 ρ18 2 2 -2 -2 0 0 0 2 2 -2 0 0 0 -2 -2 2 0 0 0 -2 2 0 0 0 0 -2 0 2 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 0 0 0 -1 -2 -2 -2 2 0 -1 -1 -1 √-3 -√-3 0 0 0 0 1 -1 -1 1 1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ20 2 2 2 2 0 0 0 -1 -2 -2 2 -2 0 -1 -1 -1 √-3 -√-3 0 0 0 0 -1 1 1 1 -1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ21 2 2 2 2 0 0 0 -1 -2 -2 2 -2 0 -1 -1 -1 -√-3 √-3 0 0 0 0 -1 1 1 1 -1 1 -√-3 √-3 complex lifted from C3⋊D4 ρ22 2 2 2 2 0 0 0 -1 -2 -2 -2 2 0 -1 -1 -1 -√-3 √-3 0 0 0 0 1 -1 -1 1 1 1 √-3 -√-3 complex lifted from C3⋊D4 ρ23 4 -4 -4 4 0 0 0 4 0 0 0 0 0 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ24 4 -4 4 -4 0 0 0 4 0 0 0 0 0 -4 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ25 4 4 -4 -4 0 0 0 -2 -4 4 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 -2 0 2 0 0 orthogonal lifted from S3×D4 ρ26 4 4 -4 -4 0 0 0 -2 4 -4 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 0 2 0 -2 0 0 orthogonal lifted from S3×D4 ρ27 4 -4 4 -4 0 0 0 -2 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 2√3 -2√3 0 0 0 0 0 orthogonal lifted from D4⋊D6 ρ28 4 -4 -4 4 0 0 0 -2 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 2√3 0 0 0 -2√3 0 0 0 orthogonal lifted from D4⋊D6 ρ29 4 -4 4 -4 0 0 0 -2 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 0 -2√3 2√3 0 0 0 0 0 orthogonal lifted from D4⋊D6 ρ30 4 -4 -4 4 0 0 0 -2 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 -2√3 0 0 0 2√3 0 0 0 orthogonal lifted from D4⋊D6

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

Matrix representation of C42.64D6 in GL6(𝔽73)

 32 3 0 0 0 0 72 41 0 0 0 0 0 0 0 0 66 59 0 0 0 0 14 7 0 0 7 14 0 0 0 0 59 66 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 72 0 0 0 0 0 0 72 0 0
,
 1 0 0 0 0 0 3 72 0 0 0 0 0 0 21 33 33 21 0 0 40 61 52 12 0 0 33 21 52 40 0 0 52 12 33 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 33 21 52 40 0 0 61 40 61 21 0 0 21 33 33 21 0 0 12 52 61 40

`G:=sub<GL(6,GF(73))| [32,72,0,0,0,0,3,41,0,0,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,66,14,0,0,0,0,59,7,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,1,0,0],[1,3,0,0,0,0,0,72,0,0,0,0,0,0,21,40,33,52,0,0,33,61,21,12,0,0,33,52,52,33,0,0,21,12,40,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,61,21,12,0,0,21,40,33,52,0,0,52,61,33,61,0,0,40,21,21,40] >;`

C42.64D6 in GAP, Magma, Sage, TeX

`C_4^2._{64}D_6`
`% in TeX`

`G:=Group("C4^2.64D6");`
`// GroupNames label`

`G:=SmallGroup(192,617);`
`// by ID`

`G=gap.SmallGroup(192,617);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,555,1123,297,136,6278]);`
`// Polycyclic`

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

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