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## G = D6⋊4S32order 432 = 24·33

### 1st 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⋊4S32
 Chief series C1 — C3 — C32 — C33 — C32×C6 — S3×C3×C6 — S32×C6 — D6⋊4S32
 Lower central C33 — C32×C6 — D6⋊4S32
 Upper central C1 — C2

Generators and relations for D64S32
G = < a,b,c,d,e,f | a6=b2=c3=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, dbd=fbf=a3b, be=eb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 2012 in 290 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×6], C3, C3 [×2], C3 [×4], C4 [×2], C22 [×9], S3 [×18], C6, C6 [×2], C6 [×16], C2×C4, D4 [×4], C23 [×2], C32, C32 [×2], C32 [×4], Dic3 [×6], C12 [×2], D6, D6 [×2], D6 [×24], C2×C6 [×12], C2×D4, C3×S3 [×16], C3⋊S3 [×2], C3⋊S3 [×9], C3×C6, C3×C6 [×2], C3×C6 [×7], C4×S3 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×10], C3×D4 [×2], C22×S3 [×6], C22×C6, C33, C3×Dic3 [×6], C3⋊Dic3 [×2], S32 [×10], S3×C6 [×6], S3×C6 [×10], C2×C3⋊S3, C2×C3⋊S3 [×9], C62 [×3], S3×D4 [×2], C2×C3⋊D4, S3×C32 [×3], C3×C3⋊S3 [×2], C33⋊C2, C32×C6, S3×Dic3 [×2], C6.D6, D6⋊S3 [×2], C3⋊D12 [×6], C3×C3⋊D4 [×4], C327D4 [×2], C2×S32, C2×S32 [×3], S3×C2×C6 [×2], C22×C3⋊S3, C3×C3⋊Dic3 [×2], C3×S32 [×2], S3×C3⋊S3 [×2], S3×C3×C6, S3×C3×C6 [×2], C6×C3⋊S3, C2×C33⋊C2, S3×C3⋊D4 [×2], Dic3⋊D6, C3×D6⋊S3 [×2], C337D4 [×2], C339(C2×C4), S32×C6, C2×S3×C3⋊S3, D64S32
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], S3×C3⋊D4 [×2], Dic3⋊D6, S33, D64S32

Permutation representations of D64S32
On 24 points - transitive group 24T1299
Generators in S24
```(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)```

`G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)>;`

`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), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16) );`

`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)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,19),(14,24),(15,23),(16,22),(17,21),(18,20)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16)])`

`G:=TransitiveGroup(24,1299);`

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 ··· 6U 6V 6W 12A 12B 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 12 12 size 1 1 6 6 6 9 9 54 2 2 2 4 4 4 8 18 18 2 2 2 4 4 4 6 6 6 6 8 12 ··· 12 18 18 36 36

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 8 8 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C3⋊D4 S32 S3×D4 C2×S32 S3×C3⋊D4 Dic3⋊D6 S33 D6⋊4S32 kernel D6⋊4S32 C3×D6⋊S3 C33⋊7D4 C33⋊9(C2×C4) S32×C6 C2×S3×C3⋊S3 D6⋊S3 C2×S32 C3×C3⋊S3 C3⋊Dic3 S3×C6 C2×C3⋊S3 C3⋊S3 D6 C32 C6 C3 C3 C2 C1 # reps 1 2 2 1 1 1 2 1 2 2 6 1 4 3 2 3 4 2 1 1

Matrix representation of D64S32 in GL8(ℤ)

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

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

D64S32 in GAP, Magma, Sage, TeX

`D_6\rtimes_4S_3^2`
`% in TeX`

`G:=Group("D6:4S3^2");`
`// GroupNames label`

`G:=SmallGroup(432,599);`
`// by ID`

`G=gap.SmallGroup(432,599);`
`# by ID`

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

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

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