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## G = C32⋊D6⋊C4order 432 = 24·33

### The semidirect product of C32⋊D6 and C4 acting via C4/C2=C2

Aliases: C32⋊D6⋊C4, C6.5S3≀C2, He3⋊(C22⋊C4), C2.2(He3⋊D4), (C2×He3).5D4, He3⋊C2.3D4, C3.(S32⋊C4), (C2×He3⋊C4)⋊1C2, He3⋊(C2×C4)⋊4C2, (C2×C32⋊D6).C2, He3⋊C2.4(C2×C4), (C2×He3⋊C2).2C22, SmallGroup(432,238)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — He3 — He3⋊C2 — C32⋊D6⋊C4
 Chief series C1 — C3 — He3 — He3⋊C2 — C2×He3⋊C2 — C2×C32⋊D6 — C32⋊D6⋊C4
 Lower central He3 — He3⋊C2 — C32⋊D6⋊C4
 Upper central C1 — C2

Generators and relations for C32⋊D6⋊C4
G = < a,b,c,d,e | a3=b3=c6=d2=e4=1, ab=ba, cac-1=dad=a-1b-1, eae-1=c4, cbc-1=ebe-1=b-1, bd=db, dcd=c-1, ece-1=ac-1d, ede-1=ac2d >

Subgroups: 907 in 99 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2 [×4], C3, C3 [×2], C4 [×2], C22 [×5], S3 [×8], C6, C6 [×6], C2×C4 [×2], C23, C32 [×2], Dic3 [×2], C12 [×2], D6 [×11], C2×C6 [×2], C22⋊C4, C3×S3 [×6], C3⋊S3 [×2], C3×C6 [×2], C4×S3, C2×Dic3, C2×C12, C22×S3 [×2], He3, C3×Dic3, C3⋊Dic3, S32 [×4], S3×C6 [×3], C2×C3⋊S3, D6⋊C4, C32⋊C6 [×2], He3⋊C2 [×2], C2×He3, S3×Dic3, C2×S32, C32⋊C12, He3⋊C4, C32⋊D6 [×2], C32⋊D6, C2×C32⋊C6, C2×He3⋊C2, He3⋊(C2×C4), C2×He3⋊C4, C2×C32⋊D6, C32⋊D6⋊C4
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, S3≀C2, S32⋊C4, He3⋊D4, C32⋊D6⋊C4

Character table of C32⋊D6⋊C4

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C 4D 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E 12F size 1 1 9 9 18 18 2 12 12 18 18 18 18 2 12 12 18 18 36 36 18 18 18 18 36 36 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 1 1 -1 -1 1 1 1 1 -1 1 -1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ5 1 -1 -1 1 1 -1 1 1 1 -i -i i i -1 -1 -1 1 -1 -1 1 -i i -i i i -i linear of order 4 ρ6 1 -1 -1 1 -1 1 1 1 1 i -i -i i -1 -1 -1 1 -1 1 -1 i -i i -i i -i linear of order 4 ρ7 1 -1 -1 1 -1 1 1 1 1 -i i i -i -1 -1 -1 1 -1 1 -1 -i i -i i -i i linear of order 4 ρ8 1 -1 -1 1 1 -1 1 1 1 i i -i -i -1 -1 -1 1 -1 -1 1 i -i i -i -i i linear of order 4 ρ9 2 -2 2 -2 0 0 2 2 2 0 0 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 -2 -2 0 0 2 2 2 0 0 0 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 4 4 0 0 0 0 4 -2 1 0 2 0 2 4 1 -2 0 0 0 0 0 0 0 0 -1 -1 orthogonal lifted from S3≀C2 ρ12 4 -4 0 0 -2 2 4 1 -2 0 0 0 0 -4 2 -1 0 0 -1 1 0 0 0 0 0 0 orthogonal lifted from S32⋊C4 ρ13 4 4 0 0 2 2 4 1 -2 0 0 0 0 4 -2 1 0 0 -1 -1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ14 4 -4 0 0 2 -2 4 1 -2 0 0 0 0 -4 2 -1 0 0 1 -1 0 0 0 0 0 0 orthogonal lifted from S32⋊C4 ρ15 4 4 0 0 0 0 4 -2 1 0 -2 0 -2 4 1 -2 0 0 0 0 0 0 0 0 1 1 orthogonal lifted from S3≀C2 ρ16 4 4 0 0 -2 -2 4 1 -2 0 0 0 0 4 -2 1 0 0 1 1 0 0 0 0 0 0 orthogonal lifted from S3≀C2 ρ17 4 -4 0 0 0 0 4 -2 1 0 2i 0 -2i -4 -1 2 0 0 0 0 0 0 0 0 i -i complex lifted from S32⋊C4 ρ18 4 -4 0 0 0 0 4 -2 1 0 -2i 0 2i -4 -1 2 0 0 0 0 0 0 0 0 -i i complex lifted from S32⋊C4 ρ19 6 6 -2 -2 0 0 -3 0 0 2 0 2 0 -3 0 0 1 1 0 0 -1 -1 -1 -1 0 0 orthogonal lifted from He3⋊D4 ρ20 6 6 -2 -2 0 0 -3 0 0 -2 0 -2 0 -3 0 0 1 1 0 0 1 1 1 1 0 0 orthogonal lifted from He3⋊D4 ρ21 6 6 2 2 0 0 -3 0 0 0 0 0 0 -3 0 0 -1 -1 0 0 √3 -√3 -√3 √3 0 0 orthogonal lifted from He3⋊D4 ρ22 6 6 2 2 0 0 -3 0 0 0 0 0 0 -3 0 0 -1 -1 0 0 -√3 √3 √3 -√3 0 0 orthogonal lifted from He3⋊D4 ρ23 6 -6 2 -2 0 0 -3 0 0 -2i 0 2i 0 3 0 0 1 -1 0 0 i -i i -i 0 0 complex faithful ρ24 6 -6 2 -2 0 0 -3 0 0 2i 0 -2i 0 3 0 0 1 -1 0 0 -i i -i i 0 0 complex faithful ρ25 6 -6 -2 2 0 0 -3 0 0 0 0 0 0 3 0 0 -1 1 0 0 √-3 √-3 -√-3 -√-3 0 0 complex faithful ρ26 6 -6 -2 2 0 0 -3 0 0 0 0 0 0 3 0 0 -1 1 0 0 -√-3 -√-3 √-3 √-3 0 0 complex faithful

Smallest permutation representation of C32⋊D6⋊C4
On 36 points
Generators in S36
```(1 36 33)(2 13 16)(3 32 31)(4 34 35)(5 17 18)(6 15 14)(7 25 28)(8 26 29)(10 23 20)(11 24 21)
(1 3 4)(2 6 5)(7 25 28)(8 29 26)(9 27 30)(10 23 20)(11 21 24)(12 19 22)(13 15 17)(14 18 16)(31 35 33)(32 34 36)
(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)
(8 9)(11 12)(13 18)(14 17)(15 16)(19 21)(22 24)(26 30)(27 29)(31 34)(32 33)(35 36)
(1 12 2 9)(3 22 6 30)(4 19 5 27)(7 36 10 13)(8 33 11 16)(14 26 31 24)(15 28 32 20)(17 25 34 23)(18 29 35 21)```

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

`G:=Group( (1,36,33)(2,13,16)(3,32,31)(4,34,35)(5,17,18)(6,15,14)(7,25,28)(8,26,29)(10,23,20)(11,24,21), (1,3,4)(2,6,5)(7,25,28)(8,29,26)(9,27,30)(10,23,20)(11,21,24)(12,19,22)(13,15,17)(14,18,16)(31,35,33)(32,34,36), (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), (8,9)(11,12)(13,18)(14,17)(15,16)(19,21)(22,24)(26,30)(27,29)(31,34)(32,33)(35,36), (1,12,2,9)(3,22,6,30)(4,19,5,27)(7,36,10,13)(8,33,11,16)(14,26,31,24)(15,28,32,20)(17,25,34,23)(18,29,35,21) );`

`G=PermutationGroup([(1,36,33),(2,13,16),(3,32,31),(4,34,35),(5,17,18),(6,15,14),(7,25,28),(8,26,29),(10,23,20),(11,24,21)], [(1,3,4),(2,6,5),(7,25,28),(8,29,26),(9,27,30),(10,23,20),(11,21,24),(12,19,22),(13,15,17),(14,18,16),(31,35,33),(32,34,36)], [(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)], [(8,9),(11,12),(13,18),(14,17),(15,16),(19,21),(22,24),(26,30),(27,29),(31,34),(32,33),(35,36)], [(1,12,2,9),(3,22,6,30),(4,19,5,27),(7,36,10,13),(8,33,11,16),(14,26,31,24),(15,28,32,20),(17,25,34,23),(18,29,35,21)])`

Matrix representation of C32⋊D6⋊C4 in GL6(𝔽13)

 0 0 0 0 12 1 1 1 0 0 11 12 0 0 0 0 12 0 1 0 0 0 12 0 0 0 1 0 12 0 0 0 0 1 12 0
,
 0 12 0 0 0 0 1 12 0 0 0 0 1 0 12 12 0 0 0 12 1 0 0 0 1 0 0 0 12 12 0 12 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 12 12 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 12 12 0 0 0 0 1 0 0 0 0 1 0 0 1 1 12 12 0 0
,
 4 6 9 11 9 11 2 0 11 2 11 2 2 2 9 11 11 2 0 2 11 2 0 0 2 2 11 2 9 11 0 2 0 0 11 2

`G:=sub<GL(6,GF(13))| [0,1,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,11,12,12,12,12,1,12,0,0,0,0],[0,1,1,0,1,0,12,12,0,12,0,12,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,1,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,12,0,0,0,0,0,12,1,0,0],[1,0,1,0,0,1,0,1,1,0,0,1,0,0,0,0,0,12,0,0,0,0,1,12,0,0,12,1,0,0,0,0,12,0,0,0],[4,2,2,0,2,0,6,0,2,2,2,2,9,11,9,11,11,0,11,2,11,2,2,0,9,11,11,0,9,11,11,2,2,0,11,2] >;`

C32⋊D6⋊C4 in GAP, Magma, Sage, TeX

`C_3^2\rtimes D_6\rtimes C_4`
`% in TeX`

`G:=Group("C3^2:D6:C4");`
`// GroupNames label`

`G:=SmallGroup(432,238);`
`// by ID`

`G=gap.SmallGroup(432,238);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,3,-3,85,64,1124,851,298,348,1027,537,14118,7069]);`
`// Polycyclic`

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

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