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G = C42.214D6order 192 = 26·3

34th non-split extension by C42 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C42.214D6
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C42⋊7S3 — C42.214D6
 Lower central C3 — C6 — C2×C12 — C42.214D6
 Upper central C1 — C22 — C42 — C4.4D4

Generators and relations for C42.214D6
G = < a,b,c,d | a4=b4=c6=1, d2=cbc-1=b-1, ab=ba, cac-1=a-1b2, ad=da, bd=db, dcd-1=b-1c-1 >

Subgroups: 400 in 130 conjugacy classes, 43 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3, C6, C6 [×2], C6, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×4], Q8 [×4], C23 [×2], Dic3, C12 [×2], C12 [×3], D6 [×3], C2×C6, C2×C6 [×3], C42, C22⋊C4 [×4], C2×C8 [×2], D8 [×2], SD16 [×4], Q16 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8, C3⋊C8 [×4], Dic6 [×2], D12 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C2×C12, C3×D4 [×2], C3×Q8 [×2], C22×S3, C22×C6, C4×C8, C4.4D4, C4.4D4, C2×D8, C2×SD16 [×2], C2×Q16, C2×C3⋊C8 [×2], D6⋊C4 [×2], D4⋊S3 [×2], D4.S3 [×2], Q82S3 [×2], C3⋊Q16 [×2], C4×C12, C3×C22⋊C4 [×2], C2×Dic6, C2×D12, C6×D4, C6×Q8, C8.12D4, C4×C3⋊C8, C427S3, C2×D4⋊S3, C2×D4.S3, C2×Q82S3, C2×C3⋊Q16, C3×C4.4D4, C42.214D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C4○D8 [×2], S3×D4 [×2], C2×C3⋊D4, C8.12D4, C123D4, Q8.13D6 [×2], C42.214D6

Smallest permutation representation of C42.214D6
On 96 points
Generators in S96
```(1 28 82 90)(2 29 83 91)(3 30 84 92)(4 31 85 93)(5 32 86 94)(6 25 87 95)(7 26 88 96)(8 27 81 89)(9 68 44 75)(10 69 45 76)(11 70 46 77)(12 71 47 78)(13 72 48 79)(14 65 41 80)(15 66 42 73)(16 67 43 74)(17 33 49 59)(18 34 50 60)(19 35 51 61)(20 36 52 62)(21 37 53 63)(22 38 54 64)(23 39 55 57)(24 40 56 58)
(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 19 72 65 18 2)(3 17 66 71 20 8)(4 7 21 70 67 24)(5 23 68 69 22 6)(9 41 64 29 94 35)(10 34 95 28 57 48)(11 47 58 27 96 33)(12 40 89 26 59 46)(13 45 60 25 90 39)(14 38 91 32 61 44)(15 43 62 31 92 37)(16 36 93 30 63 42)(49 73 78 52 81 84)(50 83 82 51 79 80)(53 77 74 56 85 88)(54 87 86 55 75 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)| (1,28,82,90)(2,29,83,91)(3,30,84,92)(4,31,85,93)(5,32,86,94)(6,25,87,95)(7,26,88,96)(8,27,81,89)(9,68,44,75)(10,69,45,76)(11,70,46,77)(12,71,47,78)(13,72,48,79)(14,65,41,80)(15,66,42,73)(16,67,43,74)(17,33,49,59)(18,34,50,60)(19,35,51,61)(20,36,52,62)(21,37,53,63)(22,38,54,64)(23,39,55,57)(24,40,56,58), (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), (1,19,72,65,18,2)(3,17,66,71,20,8)(4,7,21,70,67,24)(5,23,68,69,22,6)(9,41,64,29,94,35)(10,34,95,28,57,48)(11,47,58,27,96,33)(12,40,89,26,59,46)(13,45,60,25,90,39)(14,38,91,32,61,44)(15,43,62,31,92,37)(16,36,93,30,63,42)(49,73,78,52,81,84)(50,83,82,51,79,80)(53,77,74,56,85,88)(54,87,86,55,75,76), (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( (1,28,82,90)(2,29,83,91)(3,30,84,92)(4,31,85,93)(5,32,86,94)(6,25,87,95)(7,26,88,96)(8,27,81,89)(9,68,44,75)(10,69,45,76)(11,70,46,77)(12,71,47,78)(13,72,48,79)(14,65,41,80)(15,66,42,73)(16,67,43,74)(17,33,49,59)(18,34,50,60)(19,35,51,61)(20,36,52,62)(21,37,53,63)(22,38,54,64)(23,39,55,57)(24,40,56,58), (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), (1,19,72,65,18,2)(3,17,66,71,20,8)(4,7,21,70,67,24)(5,23,68,69,22,6)(9,41,64,29,94,35)(10,34,95,28,57,48)(11,47,58,27,96,33)(12,40,89,26,59,46)(13,45,60,25,90,39)(14,38,91,32,61,44)(15,43,62,31,92,37)(16,36,93,30,63,42)(49,73,78,52,81,84)(50,83,82,51,79,80)(53,77,74,56,85,88)(54,87,86,55,75,76), (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([(1,28,82,90),(2,29,83,91),(3,30,84,92),(4,31,85,93),(5,32,86,94),(6,25,87,95),(7,26,88,96),(8,27,81,89),(9,68,44,75),(10,69,45,76),(11,70,46,77),(12,71,47,78),(13,72,48,79),(14,65,41,80),(15,66,42,73),(16,67,43,74),(17,33,49,59),(18,34,50,60),(19,35,51,61),(20,36,52,62),(21,37,53,63),(22,38,54,64),(23,39,55,57),(24,40,56,58)], [(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)], [(1,19,72,65,18,2),(3,17,66,71,20,8),(4,7,21,70,67,24),(5,23,68,69,22,6),(9,41,64,29,94,35),(10,34,95,28,57,48),(11,47,58,27,96,33),(12,40,89,26,59,46),(13,45,60,25,90,39),(14,38,91,32,61,44),(15,43,62,31,92,37),(16,36,93,30,63,42),(49,73,78,52,81,84),(50,83,82,51,79,80),(53,77,74,56,85,88),(54,87,86,55,75,76)], [(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)])`

36 conjugacy classes

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

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C3⋊D4 C4○D8 S3×D4 Q8.13D6 kernel C42.214D6 C4×C3⋊C8 C42⋊7S3 C2×D4⋊S3 C2×D4.S3 C2×Q8⋊2S3 C2×C3⋊Q16 C3×C4.4D4 C4.4D4 C3⋊C8 C2×C12 C42 C2×D4 C2×Q8 C2×C4 C6 C4 C2 # reps 1 1 1 1 1 1 1 1 1 4 2 1 1 1 4 8 2 4

Matrix representation of C42.214D6 in GL6(𝔽73)

 27 0 0 0 0 0 0 27 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 71 0 0 0 0 37 0
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 16 16 0 0 0 0 16 57 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 2 0 0 0 0 37 0
,
 16 57 0 0 0 0 16 16 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 71 0 0 0 0 37 0

`G:=sub<GL(6,GF(73))| [27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,37,0,0,0,0,71,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[16,16,0,0,0,0,16,57,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,37,0,0,0,0,2,0],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,37,0,0,0,0,71,0] >;`

C42.214D6 in GAP, Magma, Sage, TeX

`C_4^2._{214}D_6`
`% in TeX`

`G:=Group("C4^2.214D6");`
`// GroupNames label`

`G:=SmallGroup(192,618);`
`// by ID`

`G=gap.SmallGroup(192,618);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,219,1123,297,136,6278]);`
`// 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,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^-1>;`
`// generators/relations`

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