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## G = C32⋊4D6order 108 = 22·33

### The semidirect product of C32 and D6 acting via D6/C3=C22

Aliases: C324D6, C333C22, C32S32, C3⋊S32S3, (C3×C3⋊S3)⋊3C2, SmallGroup(108,40)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C32⋊4D6
 Chief series C1 — C3 — C32 — C33 — C3×C3⋊S3 — C32⋊4D6
 Lower central C33 — C32⋊4D6
 Upper central C1

Generators and relations for C324D6
G = < a,b,c,d | a3=b3=c6=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 224 in 54 conjugacy classes, 15 normal (3 characteristic)
C1, C2 [×3], C3 [×3], C3 [×4], C22, S3 [×9], C6 [×3], C32 [×3], C32 [×4], D6 [×3], C3×S3 [×9], C3⋊S3 [×3], C33, S32 [×3], C3×C3⋊S3 [×3], C324D6
Quotients: C1, C2 [×3], C22, S3 [×3], D6 [×3], S32 [×3], C324D6

Character table of C324D6

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C size 1 9 9 9 2 2 2 4 4 4 4 4 18 18 18 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 2 2 0 0 2 2 -1 -1 -1 -1 -1 2 0 0 -1 orthogonal lifted from S3 ρ6 2 0 -2 0 -1 2 2 -1 -1 -1 2 -1 0 1 0 orthogonal lifted from D6 ρ7 2 -2 0 0 2 2 -1 -1 -1 -1 -1 2 0 0 1 orthogonal lifted from D6 ρ8 2 0 2 0 -1 2 2 -1 -1 -1 2 -1 0 -1 0 orthogonal lifted from S3 ρ9 2 0 0 2 2 -1 2 -1 -1 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ10 2 0 0 -2 2 -1 2 -1 -1 2 -1 -1 1 0 0 orthogonal lifted from D6 ρ11 4 0 0 0 4 -2 -2 1 1 -2 1 -2 0 0 0 orthogonal lifted from S32 ρ12 4 0 0 0 -2 4 -2 1 1 1 -2 -2 0 0 0 orthogonal lifted from S32 ρ13 4 0 0 0 -2 -2 4 1 1 -2 -2 1 0 0 0 orthogonal lifted from S32 ρ14 4 0 0 0 -2 -2 -2 -1-3√-3/2 -1+3√-3/2 1 1 1 0 0 0 complex faithful ρ15 4 0 0 0 -2 -2 -2 -1+3√-3/2 -1-3√-3/2 1 1 1 0 0 0 complex faithful

Permutation representations of C324D6
On 12 points - transitive group 12T71
Generators in S12
```(1 5 3)(2 4 6)(7 11 9)(8 10 12)
(1 5 3)(2 4 6)(7 9 11)(8 12 10)
(1 2 3 4 5 6)(7 8 9 10 11 12)
(1 10)(2 9)(3 8)(4 7)(5 12)(6 11)```

`G:=sub<Sym(12)| (1,5,3)(2,4,6)(7,11,9)(8,10,12), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11)>;`

`G:=Group( (1,5,3)(2,4,6)(7,11,9)(8,10,12), (1,5,3)(2,4,6)(7,9,11)(8,12,10), (1,2,3,4,5,6)(7,8,9,10,11,12), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11) );`

`G=PermutationGroup([(1,5,3),(2,4,6),(7,11,9),(8,10,12)], [(1,5,3),(2,4,6),(7,9,11),(8,12,10)], [(1,2,3,4,5,6),(7,8,9,10,11,12)], [(1,10),(2,9),(3,8),(4,7),(5,12),(6,11)])`

`G:=TransitiveGroup(12,71);`

On 18 points - transitive group 18T53
Generators in S18
```(1 10 7)(2 8 11)(3 12 9)(4 18 15)(5 16 13)(6 14 17)
(1 7 10)(2 11 8)(3 9 12)(4 18 15)(5 16 13)(6 14 17)
(1 2 3)(4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 5)(2 4)(3 6)(7 13)(8 18)(9 17)(10 16)(11 15)(12 14)```

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

`G:=Group( (1,10,7)(2,8,11)(3,12,9)(4,18,15)(5,16,13)(6,14,17), (1,7,10)(2,11,8)(3,9,12)(4,18,15)(5,16,13)(6,14,17), (1,2,3)(4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,5)(2,4)(3,6)(7,13)(8,18)(9,17)(10,16)(11,15)(12,14) );`

`G=PermutationGroup([(1,10,7),(2,8,11),(3,12,9),(4,18,15),(5,16,13),(6,14,17)], [(1,7,10),(2,11,8),(3,9,12),(4,18,15),(5,16,13),(6,14,17)], [(1,2,3),(4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,5),(2,4),(3,6),(7,13),(8,18),(9,17),(10,16),(11,15),(12,14)])`

`G:=TransitiveGroup(18,53);`

On 27 points - transitive group 27T35
Generators in S27
```(1 11 14)(2 15 12)(3 13 10)(4 22 18)(5 19 23)(6 24 20)(7 21 25)(8 26 16)(9 17 27)
(1 7 4)(2 5 8)(3 9 6)(10 27 20)(11 21 22)(12 23 16)(13 17 24)(14 25 18)(15 19 26)
(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)
(1 3)(4 9)(5 8)(6 7)(10 14)(11 13)(16 23)(17 22)(18 27)(19 26)(20 25)(21 24)```

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

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

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

`G:=TransitiveGroup(27,35);`

C324D6 is a maximal subgroup of
C33⋊D4  C322D12  S33  C32⋊D18  He35D6  C325D18  C33⋊A4  C3317D6  C6210D6
C324D6 is a maximal quotient of
C339(C2×C4)  C339D4  C335Q8  C325D18  He36D6  He3.6D6  C3317D6  C6210D6

Polynomial with Galois group C324D6 over ℚ
actionf(x)Disc(f)
12T71x12-5x9+11x6-10x3+4210·318·56

Matrix representation of C324D6 in GL4(𝔽7) generated by

 5 3 2 3 1 3 3 0 4 4 0 6 0 0 0 4
,
 3 6 3 2 6 3 4 2 0 0 2 0 0 0 0 4
,
 5 2 2 1 0 6 1 1 2 5 6 2 5 5 2 4
,
 5 2 0 1 3 3 3 6 3 4 2 6 5 5 1 4
`G:=sub<GL(4,GF(7))| [5,1,4,0,3,3,4,0,2,3,0,0,3,0,6,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[5,0,2,5,2,6,5,5,2,1,6,2,1,1,2,4],[5,3,3,5,2,3,4,5,0,3,2,1,1,6,6,4] >;`

C324D6 in GAP, Magma, Sage, TeX

`C_3^2\rtimes_4D_6`
`% in TeX`

`G:=Group("C3^2:4D6");`
`// GroupNames label`

`G:=SmallGroup(108,40);`
`// by ID`

`G=gap.SmallGroup(108,40);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-3,-3,122,67,248,1804]);`
`// Polycyclic`

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

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