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## G = C33⋊9D4order 216 = 23·33

### 6th semidirect product of C33 and D4 acting via D4/C2=C22

Aliases: C339D4, C326D12, C6.23S32, C3⋊Dic36S3, (C3×C6).36D6, C31(D6⋊S3), C33(C3⋊D12), C328(C3⋊D4), C2.2(C324D6), (C32×C6).14C22, (C2×C3⋊S3)⋊6S3, (C6×C3⋊S3)⋊4C2, (C3×C3⋊Dic3)⋊3C2, SmallGroup(216,132)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C6 — C33⋊9D4
 Chief series C1 — C3 — C32 — C33 — C32×C6 — C6×C3⋊S3 — C33⋊9D4
 Lower central C33 — C32×C6 — C33⋊9D4
 Upper central C1 — C2

Generators and relations for C339D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 428 in 94 conjugacy classes, 23 normal (11 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C3 [×4], C4, C22 [×2], S3 [×8], C6, C6 [×2], C6 [×6], D4, C32, C32 [×2], C32 [×4], Dic3 [×3], C12, D6 [×6], C2×C6 [×2], C3×S3 [×8], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×4], D12, C3⋊D4 [×2], C33, C3×Dic3 [×3], C3⋊Dic3, S3×C6 [×6], C2×C3⋊S3 [×2], C3×C3⋊S3 [×2], C32×C6, D6⋊S3, C3⋊D12 [×2], C3×C3⋊Dic3, C6×C3⋊S3 [×2], C339D4
Quotients: C1, C2 [×3], C22, S3 [×3], D4, D6 [×3], D12, C3⋊D4 [×2], S32 [×3], D6⋊S3, C3⋊D12 [×2], C324D6, C339D4

Character table of C339D4

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 3F 3G 3H 4 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 12A 12B size 1 1 18 18 2 2 2 4 4 4 4 4 18 2 2 2 4 4 4 4 4 18 18 18 18 18 18 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 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 -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 -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 1 linear of order 2 ρ5 2 2 0 -2 -1 2 2 2 -1 -1 -1 -1 0 2 -1 2 -1 -1 -1 -1 2 1 0 0 1 0 0 orthogonal lifted from D6 ρ6 2 2 2 0 2 -1 2 -1 -1 2 -1 -1 0 2 2 -1 -1 -1 -1 2 -1 0 -1 -1 0 0 0 orthogonal lifted from S3 ρ7 2 2 0 0 2 2 -1 -1 2 -1 -1 -1 -2 -1 2 2 -1 -1 2 -1 -1 0 0 0 0 1 1 orthogonal lifted from D6 ρ8 2 2 -2 0 2 -1 2 -1 -1 2 -1 -1 0 2 2 -1 -1 -1 -1 2 -1 0 1 1 0 0 0 orthogonal lifted from D6 ρ9 2 2 0 2 -1 2 2 2 -1 -1 -1 -1 0 2 -1 2 -1 -1 -1 -1 2 -1 0 0 -1 0 0 orthogonal lifted from S3 ρ10 2 2 0 0 2 2 -1 -1 2 -1 -1 -1 2 -1 2 2 -1 -1 2 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ11 2 -2 0 0 2 2 2 2 2 2 2 2 0 -2 -2 -2 -2 -2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 -2 0 0 2 2 -1 -1 2 -1 -1 -1 0 1 -2 -2 1 1 -2 1 1 0 0 0 0 -√3 √3 orthogonal lifted from D12 ρ13 2 -2 0 0 2 2 -1 -1 2 -1 -1 -1 0 1 -2 -2 1 1 -2 1 1 0 0 0 0 √3 -√3 orthogonal lifted from D12 ρ14 2 -2 0 0 2 -1 2 -1 -1 2 -1 -1 0 -2 -2 1 1 1 1 -2 1 0 √-3 -√-3 0 0 0 complex lifted from C3⋊D4 ρ15 2 -2 0 0 -1 2 2 2 -1 -1 -1 -1 0 -2 1 -2 1 1 1 1 -2 -√-3 0 0 √-3 0 0 complex lifted from C3⋊D4 ρ16 2 -2 0 0 -1 2 2 2 -1 -1 -1 -1 0 -2 1 -2 1 1 1 1 -2 √-3 0 0 -√-3 0 0 complex lifted from C3⋊D4 ρ17 2 -2 0 0 2 -1 2 -1 -1 2 -1 -1 0 -2 -2 1 1 1 1 -2 1 0 -√-3 √-3 0 0 0 complex lifted from C3⋊D4 ρ18 4 4 0 0 -2 -2 4 -2 1 -2 1 1 0 4 -2 -2 1 1 1 -2 -2 0 0 0 0 0 0 orthogonal lifted from S32 ρ19 4 4 0 0 4 -2 -2 1 -2 -2 1 1 0 -2 4 -2 1 1 -2 -2 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ20 4 4 0 0 -2 4 -2 -2 -2 1 1 1 0 -2 -2 4 1 1 -2 1 -2 0 0 0 0 0 0 orthogonal lifted from S32 ρ21 4 -4 0 0 -2 4 -2 -2 -2 1 1 1 0 2 2 -4 -1 -1 2 -1 2 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ22 4 -4 0 0 4 -2 -2 1 -2 -2 1 1 0 2 -4 2 -1 -1 2 2 -1 0 0 0 0 0 0 orthogonal lifted from C3⋊D12 ρ23 4 -4 0 0 -2 -2 4 -2 1 -2 1 1 0 -4 2 2 -1 -1 -1 2 2 0 0 0 0 0 0 symplectic lifted from D6⋊S3, Schur index 2 ρ24 4 -4 0 0 -2 -2 -2 1 1 1 -1+3√-3/2 -1-3√-3/2 0 2 2 2 1-3√-3/2 1+3√-3/2 -1 -1 -1 0 0 0 0 0 0 complex faithful ρ25 4 4 0 0 -2 -2 -2 1 1 1 -1+3√-3/2 -1-3√-3/2 0 -2 -2 -2 -1+3√-3/2 -1-3√-3/2 1 1 1 0 0 0 0 0 0 complex lifted from C32⋊4D6 ρ26 4 -4 0 0 -2 -2 -2 1 1 1 -1-3√-3/2 -1+3√-3/2 0 2 2 2 1+3√-3/2 1-3√-3/2 -1 -1 -1 0 0 0 0 0 0 complex faithful ρ27 4 4 0 0 -2 -2 -2 1 1 1 -1-3√-3/2 -1+3√-3/2 0 -2 -2 -2 -1-3√-3/2 -1+3√-3/2 1 1 1 0 0 0 0 0 0 complex lifted from C32⋊4D6

Permutation representations of C339D4
On 24 points - transitive group 24T550
Generators in S24
```(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 14 10)(6 15 11)(7 16 12)(8 13 9)
(1 20 21)(2 22 17)(3 18 23)(4 24 19)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(1 21 20)(2 17 22)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)```

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

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

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

`G:=TransitiveGroup(24,550);`

C339D4 is a maximal subgroup of
C33⋊D8  C336SD16  C322D24  C338SD16  S3×D6⋊S3  S3×C3⋊D12  (S3×C6)⋊D6  D6.S32  D6.6S32  Dic3.S32  C12⋊S312S3  C12.95S32  C123S32  C62.96D6  C6224D6
C339D4 is a maximal quotient of
C339D8  C3318SD16  C339Q16  C62.84D6  C62.85D6

Matrix representation of C339D4 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
,
 0 5 2 6 0 2 0 2 3 3 6 1 0 0 0 4
,
 6 5 0 5 0 0 4 3 1 6 3 5 1 1 3 5
,
 2 6 3 4 0 4 6 5 4 4 1 6 5 2 1 0
`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],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[6,0,1,1,5,0,6,1,0,4,3,3,5,3,5,5],[2,0,4,5,6,4,4,2,3,6,1,1,4,5,6,0] >;`

C339D4 in GAP, Magma, Sage, TeX

`C_3^3\rtimes_9D_4`
`% in TeX`

`G:=Group("C3^3:9D4");`
`// GroupNames label`

`G:=SmallGroup(216,132);`
`// by ID`

`G=gap.SmallGroup(216,132);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,387,201,730,5189]);`
`// Polycyclic`

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

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