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## G = C33⋊C2order 54 = 2·33

### 3rd semidirect product of C33 and C2 acting faithfully

Aliases: C333C2, C324S3, C3⋊(C3⋊S3), SmallGroup(54,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C33 — C33⋊C2
 Chief series C1 — C3 — C32 — C33 — C33⋊C2
 Lower central C33 — C33⋊C2
 Upper central C1

Generators and relations for C33⋊C2
G = < a,b,c,d | a3=b3=c3=d2=1, ab=ba, ac=ca, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 212 in 56 conjugacy classes, 29 normal (3 characteristic)
C1, C2, C3 [×13], S3 [×13], C32 [×13], C3⋊S3 [×13], C33, C33⋊C2
Quotients: C1, C2, S3 [×13], C3⋊S3 [×13], C33⋊C2

Character table of C33⋊C2

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M size 1 27 2 2 2 2 2 2 2 2 2 2 2 2 2 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 2 0 -1 -1 2 2 -1 -1 -1 -1 2 -1 2 -1 -1 orthogonal lifted from S3 ρ4 2 0 -1 2 -1 2 -1 -1 -1 2 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ5 2 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 2 2 2 orthogonal lifted from S3 ρ6 2 0 -1 2 -1 -1 -1 2 2 -1 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ7 2 0 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 -1 -1 2 -1 2 -1 2 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ9 2 0 2 -1 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 orthogonal lifted from S3 ρ10 2 0 -1 -1 -1 -1 -1 -1 2 2 2 -1 -1 -1 2 orthogonal lifted from S3 ρ11 2 0 2 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ12 2 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ13 2 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ14 2 0 -1 -1 2 -1 -1 2 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ15 2 0 -1 2 -1 -1 2 -1 -1 -1 2 2 -1 -1 -1 orthogonal lifted from S3

Permutation representations of C33⋊C2
On 27 points - transitive group 27T7
Generators in S27
```(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 6 14)(2 4 15)(3 5 13)(7 24 16)(8 22 17)(9 23 18)(10 27 19)(11 25 20)(12 26 21)
(1 26 8)(2 27 9)(3 25 7)(4 19 23)(5 20 24)(6 21 22)(10 18 15)(11 16 13)(12 17 14)
(2 3)(4 13)(5 15)(6 14)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(16 19)(17 21)(18 20)```

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

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

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

`G:=TransitiveGroup(27,7);`

C33⋊C2 is a maximal subgroup of
S3×C3⋊S3  C33⋊C6  He34S3  C324D9  C34⋊C2  C324S4  C33⋊D5  C33⋊D7
C33⋊C2 is a maximal quotient of
C335C4  C324D9  He35S3  C34⋊C2  C324S4  C33⋊D5  C33⋊D7

Matrix representation of C33⋊C2 in GL6(ℤ)

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

`G:=sub<GL(6,Integers())| [0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1] >;`

C33⋊C2 in GAP, Magma, Sage, TeX

`C_3^3\rtimes C_2`
`% in TeX`

`G:=Group("C3^3:C2");`
`// GroupNames label`

`G:=SmallGroup(54,14);`
`// by ID`

`G=gap.SmallGroup(54,14);`
`# by ID`

`G:=PCGroup([4,-2,-3,-3,-3,33,146,579]);`
`// Polycyclic`

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

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