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## G = D6⋊C42order 192 = 26·3

### 1st semidirect product of D6 and C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — D6⋊C42
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — S3×C22×C4 — D6⋊C42
 Lower central C3 — C6 — D6⋊C42
 Upper central C1 — C23 — C2.C42

Generators and relations for D6⋊C42
G = < a,b,c,d | a6=b2=c4=d4=1, bab=cac-1=a-1, ad=da, cbc-1=a4b, dbd-1=a3b, cd=dc >

Subgroups: 640 in 258 conjugacy classes, 99 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×14], C22 [×3], C22 [×4], C22 [×16], S3 [×4], C6 [×3], C6 [×4], C2×C4 [×6], C2×C4 [×28], C23, C23 [×10], Dic3 [×4], Dic3 [×4], C12 [×6], D6 [×4], D6 [×12], C2×C6 [×3], C2×C6 [×4], C42 [×4], C22⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×11], C24, C4×S3 [×8], C2×Dic3 [×10], C2×Dic3 [×4], C2×C12 [×6], C2×C12 [×6], C22×S3 [×6], C22×S3 [×4], C22×C6, C2.C42, C2.C42, C2×C42 [×2], C2×C22⋊C4 [×2], C23×C4, C4×Dic3 [×4], D6⋊C4 [×8], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23, C4×C22⋊C4, C6.C42, C3×C2.C42, C2×C4×Dic3 [×2], C2×D6⋊C4 [×2], S3×C22×C4, D6⋊C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3, C2×C4 [×18], D4 [×4], C23, D6 [×3], C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C4×S3 [×6], C22×S3, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], S3×C2×C4 [×3], S3×D4 [×2], D42S3, Q83S3, C4×C22⋊C4, S3×C42, S3×C22⋊C4, Dic34D4 [×2], C4⋊C47S3, Dic35D4 [×2], D6⋊C42

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

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4L 4M ··· 4T 4U ··· 4AB 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 6 6 6 2 2 ··· 2 3 ··· 3 6 ··· 6 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C4 C4 S3 D4 D6 C4○D4 C4×S3 S3×D4 D4⋊2S3 Q8⋊3S3 kernel D6⋊C42 C6.C42 C3×C2.C42 C2×C4×Dic3 C2×D6⋊C4 S3×C22×C4 D6⋊C4 S3×C2×C4 C2.C42 C2×Dic3 C22×C4 C2×C6 C2×C4 C22 C22 C22 # reps 1 1 1 2 2 1 16 8 1 4 3 4 12 2 1 1

Matrix representation of D6⋊C42 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 12 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;`

D6⋊C42 in GAP, Magma, Sage, TeX

`D_6\rtimes C_4^2`
`% in TeX`

`G:=Group("D6:C4^2");`
`// GroupNames label`

`G:=SmallGroup(192,225);`
`// by ID`

`G=gap.SmallGroup(192,225);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,387,58,6278]);`
`// Polycyclic`

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

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