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G = D6⋊C8⋊C2order 192 = 26·3

23rd semidirect product of D6⋊C8 and C2 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D6⋊C8⋊C2
 Chief series C1 — C3 — C6 — C12 — C2×C12 — S3×C2×C4 — C2×C4○D12 — D6⋊C8⋊C2
 Lower central C3 — C2×C6 — D6⋊C8⋊C2
 Upper central C1 — C2×C4 — C22⋊C8

Generators and relations for D6⋊C8⋊C2
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, dbd=bc4, dcd=a3c5 >

Subgroups: 416 in 158 conjugacy classes, 55 normal (47 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×4], C22, C22 [×10], S3 [×3], C6 [×3], C6, C8 [×4], C2×C4 [×2], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], Dic3 [×2], Dic3, C12 [×2], C12, D6 [×2], D6 [×5], C2×C6, C2×C6 [×3], C2×C8 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], C3⋊C8 [×2], C24 [×2], Dic6 [×2], C4×S3 [×4], C4×S3 [×2], D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C22×C6, C22⋊C8, C22⋊C8 [×3], C22×C8, C2×M4(2), C2×C4○D4, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8 [×2], C2×C24 [×2], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C2×C3⋊D4 [×2], C22×C12, (C22×C8)⋊C2, D6⋊C8 [×2], C12.55D4, C3×C22⋊C8, S3×C2×C8, C2×C8⋊S3, C2×C4○D12, D6⋊C8⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, C8○D4 [×2], S3×C2×C4, S3×D4 [×2], (C22×C8)⋊C2, S3×C22⋊C4, C8○D12, D12.C4, D6⋊C8⋊C2

Smallest permutation representation of D6⋊C8⋊C2
On 96 points
Generators in S96
```(1 63 29 17 93 73)(2 64 30 18 94 74)(3 57 31 19 95 75)(4 58 32 20 96 76)(5 59 25 21 89 77)(6 60 26 22 90 78)(7 61 27 23 91 79)(8 62 28 24 92 80)(9 85 67 52 38 42)(10 86 68 53 39 43)(11 87 69 54 40 44)(12 88 70 55 33 45)(13 81 71 56 34 46)(14 82 72 49 35 47)(15 83 65 50 36 48)(16 84 66 51 37 41)
(1 25)(2 78)(3 27)(4 80)(5 29)(6 74)(7 31)(8 76)(9 42)(10 68)(11 44)(12 70)(13 46)(14 72)(15 48)(16 66)(17 77)(18 26)(19 79)(20 28)(21 73)(22 30)(23 75)(24 32)(34 81)(36 83)(38 85)(40 87)(41 51)(43 53)(45 55)(47 49)(50 65)(52 67)(54 69)(56 71)(57 61)(58 92)(59 63)(60 94)(62 96)(64 90)(89 93)(91 95)
(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 53)(2 15)(3 55)(4 9)(5 49)(6 11)(7 51)(8 13)(10 17)(12 19)(14 21)(16 23)(18 50)(20 52)(22 54)(24 56)(25 47)(26 69)(27 41)(28 71)(29 43)(30 65)(31 45)(32 67)(33 57)(34 92)(35 59)(36 94)(37 61)(38 96)(39 63)(40 90)(42 76)(44 78)(46 80)(48 74)(58 85)(60 87)(62 81)(64 83)(66 79)(68 73)(70 75)(72 77)(82 89)(84 91)(86 93)(88 95)```

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 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 C4 C4 C4 S3 D4 D6 D6 C4×S3 C4×S3 C8○D4 C8○D12 S3×D4 D12.C4 kernel D6⋊C8⋊C2 D6⋊C8 C12.55D4 C3×C22⋊C8 S3×C2×C8 C2×C8⋊S3 C2×C4○D12 C2×Dic6 C2×D12 C2×C3⋊D4 C22⋊C8 C4×S3 C2×C8 C22×C4 C2×C4 C23 C6 C2 C4 C2 # reps 1 2 1 1 1 1 1 2 2 4 1 4 2 1 2 2 8 8 2 2

Matrix representation of D6⋊C8⋊C2 in GL4(𝔽73) generated by

 1 72 0 0 1 0 0 0 0 0 72 0 0 0 0 72
,
 0 72 0 0 72 0 0 0 0 0 1 0 0 0 46 72
,
 70 6 0 0 67 3 0 0 0 0 27 2 0 0 0 46
,
 43 60 0 0 13 30 0 0 0 0 1 0 0 0 46 72
`G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,72,0,0,0,0,72],[0,72,0,0,72,0,0,0,0,0,1,46,0,0,0,72],[70,67,0,0,6,3,0,0,0,0,27,0,0,0,2,46],[43,13,0,0,60,30,0,0,0,0,1,46,0,0,0,72] >;`

D6⋊C8⋊C2 in GAP, Magma, Sage, TeX

`D_6\rtimes C_8\rtimes C_2`
`% in TeX`

`G:=Group("D6:C8:C2");`
`// GroupNames label`

`G:=SmallGroup(192,286);`
`// by ID`

`G=gap.SmallGroup(192,286);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,219,58,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,d*b*d=b*c^4,d*c*d=a^3*c^5>;`
`// generators/relations`

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