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G = C42.56D6order 192 = 26·3

56th non-split extension by C42 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C42.56D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×Q8⋊2S3 — C42.56D6
 Lower central C3 — C6 — C12 — C42.56D6
 Upper central C1 — C22 — C42 — C4×Q8

Generators and relations for C42.56D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c5 >

Subgroups: 328 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], C6 [×3], C8 [×3], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], Q8, C23, Dic3, C12 [×2], C12 [×5], D6 [×4], C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, C3⋊C8 [×2], C3⋊C8, C4×S3 [×2], D12 [×2], D12, C2×Dic3, C2×C12 [×3], C2×C12 [×2], C3×Q8 [×2], C3×Q8, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, Q82S3 [×4], C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C6×Q8, SD16⋊C4, C42.S3, C6.Q16, C6.D8, Q82Dic3, C4×D12, C2×Q82S3, Q8×C12, C42.56D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C3⋊D4 [×2], C22×S3, C4×D4, C8⋊C22, C8.C22, S3×C2×C4, C4○D12, C2×C3⋊D4, SD16⋊C4, C4×C3⋊D4, Q8.11D6, D4⋊D6, C42.56D6

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 C8⋊C22 C8.C22 Q8.11D6 D4⋊D6 kernel C42.56D6 C42.S3 C6.Q16 C6.D8 Q8⋊2Dic3 C4×D12 C2×Q8⋊2S3 Q8×C12 Q8⋊2S3 C4×Q8 C2×C12 C42 C4⋊C4 C2×Q8 C12 C2×C4 Q8 C4 C6 C6 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 2 4 4 4 1 1 2 2

Matrix representation of C42.56D6 in GL8(𝔽73)

 46 0 0 0 0 0 0 0 0 46 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 71 0 67 0 0 0 0 0 14 46 2 71 0 0 0 0 37 0 2 0 0 0 0 0 53 72 6 27
,
 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 48 72 0 0 0 0 0 0 66 49 0 1 0 0 0 0 64 70 72 0
,
 68 67 0 0 0 0 0 0 4 5 0 0 0 0 0 0 0 0 13 30 0 0 0 0 0 0 43 43 0 0 0 0 0 0 0 0 44 60 41 41 0 0 0 0 8 33 0 70 0 0 0 0 20 56 17 17 0 0 0 0 35 61 12 52
,
 5 6 0 0 0 0 0 0 20 68 0 0 0 0 0 0 0 0 30 30 0 0 0 0 0 0 60 43 0 0 0 0 0 0 0 0 68 13 41 32 0 0 0 0 65 40 0 3 0 0 0 0 46 17 17 56 0 0 0 0 12 12 12 21

`G:=sub<GL(8,GF(73))| [46,0,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,71,14,37,53,0,0,0,0,0,46,0,72,0,0,0,0,67,2,2,6,0,0,0,0,0,71,0,27],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,48,66,64,0,0,0,0,3,72,49,70,0,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0],[68,4,0,0,0,0,0,0,67,5,0,0,0,0,0,0,0,0,13,43,0,0,0,0,0,0,30,43,0,0,0,0,0,0,0,0,44,8,20,35,0,0,0,0,60,33,56,61,0,0,0,0,41,0,17,12,0,0,0,0,41,70,17,52],[5,20,0,0,0,0,0,0,6,68,0,0,0,0,0,0,0,0,30,60,0,0,0,0,0,0,30,43,0,0,0,0,0,0,0,0,68,65,46,12,0,0,0,0,13,40,17,12,0,0,0,0,41,0,17,12,0,0,0,0,32,3,56,21] >;`

C42.56D6 in GAP, Magma, Sage, TeX

`C_4^2._{56}D_6`
`% in TeX`

`G:=Group("C4^2.56D6");`
`// GroupNames label`

`G:=SmallGroup(192,585);`
`// by ID`

`G=gap.SmallGroup(192,585);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,387,58,1684,851,102,6278]);`
`// Polycyclic`

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

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