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## G = D6⋊S32order 432 = 24·33

### 3rd semidirect product of D6 and S32 acting via S32/C3×S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — D6⋊S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S32×C6 — D6⋊S32
 Lower central C33 — C32×C6 — D6⋊S32
 Upper central C1 — C2

Generators and relations for D6⋊S32
G = < a,b,c,d,e,f | a3=b3=c3=d2=e4=f2=1, ab=ba, ac=ca, ad=da, eae-1=a-1, af=fa, bc=cb, dbd=ebe-1=fbf=b-1, dcd=ece-1=c-1, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1676 in 270 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×2], C22 [×9], S3 [×14], C6, C6 [×2], C6 [×14], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3, Dic3 [×7], C12 [×3], D6 [×2], D6 [×19], C2×C6 [×11], C2×D4, C3×S3 [×18], C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×6], C4×S3 [×3], D12 [×2], C2×Dic3, C3⋊D4 [×10], C3×D4 [×2], C22×S3 [×5], C22×C6, C33, C3×Dic3 [×2], C3×Dic3, C3⋊Dic3 [×7], C3×C12, S32 [×8], S3×C6 [×4], S3×C6 [×15], C2×C3⋊S3, C2×C3⋊S3 [×2], C62 [×2], S3×D4 [×2], C2×C3⋊D4, S3×C32 [×2], C3×C3⋊S3 [×2], C3×C3⋊S3 [×2], C32×C6, S3×Dic3 [×3], D6⋊S3 [×6], C3⋊D12 [×2], C3×D12 [×2], C3×C3⋊D4 [×2], C4×C3⋊S3, C327D4 [×2], C2×S32, C2×S32 [×3], S3×C2×C6 [×2], C32×Dic3, C335C4, C3×S32 [×2], C324D6 [×2], S3×C3×C6 [×2], C6×C3⋊S3, C6×C3⋊S3 [×2], D6⋊D6, S3×C3⋊D4 [×2], C3×C3⋊D12 [×2], Dic3×C3⋊S3, C336D4 [×2], S32×C6, C2×C324D6, D6⋊S32
Quotients: C1, C2 [×7], C22 [×7], S3 [×3], D4 [×2], C23, D6 [×9], C2×D4, C3⋊D4 [×2], C22×S3 [×3], S32 [×3], S3×D4 [×2], C2×C3⋊D4, C2×S32 [×3], D6⋊D6, S3×C3⋊D4 [×2], S33, D6⋊S32

Smallest permutation representation of D6⋊S32
On 48 points
Generators in S48
```(1 44 45)(2 46 41)(3 42 47)(4 48 43)(5 28 24)(6 21 25)(7 26 22)(8 23 27)(9 37 30)(10 31 38)(11 39 32)(12 29 40)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 44 45)(2 46 41)(3 42 47)(4 48 43)(5 24 28)(6 25 21)(7 22 26)(8 27 23)(9 30 37)(10 38 31)(11 32 39)(12 40 29)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 45 44)(2 41 46)(3 47 42)(4 43 48)(5 28 24)(6 21 25)(7 26 22)(8 23 27)(9 30 37)(10 38 31)(11 32 39)(12 40 29)(13 18 35)(14 36 19)(15 20 33)(16 34 17)
(1 6)(2 7)(3 8)(4 5)(9 19)(10 20)(11 17)(12 18)(13 40)(14 37)(15 38)(16 39)(21 44)(22 41)(23 42)(24 43)(25 45)(26 46)(27 47)(28 48)(29 35)(30 36)(31 33)(32 34)
(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)
(1 30)(2 29)(3 32)(4 31)(5 33)(6 36)(7 35)(8 34)(9 44)(10 43)(11 42)(12 41)(13 26)(14 25)(15 28)(16 27)(17 23)(18 22)(19 21)(20 24)(37 45)(38 48)(39 47)(40 46)```

`G:=sub<Sym(48)| (1,44,45)(2,46,41)(3,42,47)(4,48,43)(5,28,24)(6,21,25)(7,26,22)(8,23,27)(9,37,30)(10,31,38)(11,39,32)(12,29,40)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,44,45)(2,46,41)(3,42,47)(4,48,43)(5,24,28)(6,25,21)(7,22,26)(8,27,23)(9,30,37)(10,38,31)(11,32,39)(12,40,29)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,45,44)(2,41,46)(3,47,42)(4,43,48)(5,28,24)(6,21,25)(7,26,22)(8,23,27)(9,30,37)(10,38,31)(11,32,39)(12,40,29)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,40)(14,37)(15,38)(16,39)(21,44)(22,41)(23,42)(24,43)(25,45)(26,46)(27,47)(28,48)(29,35)(30,36)(31,33)(32,34), (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), (1,30)(2,29)(3,32)(4,31)(5,33)(6,36)(7,35)(8,34)(9,44)(10,43)(11,42)(12,41)(13,26)(14,25)(15,28)(16,27)(17,23)(18,22)(19,21)(20,24)(37,45)(38,48)(39,47)(40,46)>;`

`G:=Group( (1,44,45)(2,46,41)(3,42,47)(4,48,43)(5,28,24)(6,21,25)(7,26,22)(8,23,27)(9,37,30)(10,31,38)(11,39,32)(12,29,40)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,44,45)(2,46,41)(3,42,47)(4,48,43)(5,24,28)(6,25,21)(7,22,26)(8,27,23)(9,30,37)(10,38,31)(11,32,39)(12,40,29)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,45,44)(2,41,46)(3,47,42)(4,43,48)(5,28,24)(6,21,25)(7,26,22)(8,23,27)(9,30,37)(10,38,31)(11,32,39)(12,40,29)(13,18,35)(14,36,19)(15,20,33)(16,34,17), (1,6)(2,7)(3,8)(4,5)(9,19)(10,20)(11,17)(12,18)(13,40)(14,37)(15,38)(16,39)(21,44)(22,41)(23,42)(24,43)(25,45)(26,46)(27,47)(28,48)(29,35)(30,36)(31,33)(32,34), (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), (1,30)(2,29)(3,32)(4,31)(5,33)(6,36)(7,35)(8,34)(9,44)(10,43)(11,42)(12,41)(13,26)(14,25)(15,28)(16,27)(17,23)(18,22)(19,21)(20,24)(37,45)(38,48)(39,47)(40,46) );`

`G=PermutationGroup([(1,44,45),(2,46,41),(3,42,47),(4,48,43),(5,28,24),(6,21,25),(7,26,22),(8,23,27),(9,37,30),(10,31,38),(11,39,32),(12,29,40),(13,18,35),(14,36,19),(15,20,33),(16,34,17)], [(1,44,45),(2,46,41),(3,42,47),(4,48,43),(5,24,28),(6,25,21),(7,22,26),(8,27,23),(9,30,37),(10,38,31),(11,32,39),(12,40,29),(13,18,35),(14,36,19),(15,20,33),(16,34,17)], [(1,45,44),(2,41,46),(3,47,42),(4,43,48),(5,28,24),(6,21,25),(7,26,22),(8,23,27),(9,30,37),(10,38,31),(11,32,39),(12,40,29),(13,18,35),(14,36,19),(15,20,33),(16,34,17)], [(1,6),(2,7),(3,8),(4,5),(9,19),(10,20),(11,17),(12,18),(13,40),(14,37),(15,38),(16,39),(21,44),(22,41),(23,42),(24,43),(25,45),(26,46),(27,47),(28,48),(29,35),(30,36),(31,33),(32,34)], [(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)], [(1,30),(2,29),(3,32),(4,31),(5,33),(6,36),(7,35),(8,34),(9,44),(10,43),(11,42),(12,41),(13,26),(14,25),(15,28),(16,27),(17,23),(18,22),(19,21),(20,24),(37,45),(38,48),(39,47),(40,46)])`

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 3F 3G 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L ··· 6Q 6R 6S 6T 6U 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 6 6 6 ··· 6 6 6 6 6 12 12 12 12 size 1 1 6 6 9 9 18 18 2 2 2 4 4 4 8 6 54 2 2 2 4 4 4 6 6 6 6 8 12 ··· 12 18 18 36 36 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 8 8 type + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C3⋊D4 S32 S32 S3×D4 C2×S32 D6⋊D6 S3×C3⋊D4 S33 D6⋊S32 kernel D6⋊S32 C3×C3⋊D12 Dic3×C3⋊S3 C33⋊6D4 S32×C6 C2×C32⋊4D6 C3⋊D12 C2×S32 C3×C3⋊S3 C3×Dic3 S3×C6 C2×C3⋊S3 C3⋊S3 Dic3 D6 C32 C6 C3 C3 C2 C1 # reps 1 2 1 2 1 1 2 1 2 2 4 3 4 1 2 2 3 2 4 1 1

Matrix representation of D6⋊S32 in GL8(ℤ)

 1 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 1 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 0 -1 0 0 0 0 0 0 1 -1
,
 1 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 1 0 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 -1 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 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0
,
 -1 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 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1

`G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;`

D6⋊S32 in GAP, Magma, Sage, TeX

`D_6\rtimes S_3^2`
`% in TeX`

`G:=Group("D6:S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,600);`
`// by ID`

`G=gap.SmallGroup(432,600);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,298,2028,14118]);`
`// Polycyclic`

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

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