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G = C33order 27 = 33

Elementary abelian group of type [3,3,3]

Aliases: C33, SmallGroup(27,5)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C33
 Chief series C1 — C3 — C32 — C33
 Lower central C1 — C33
 Upper central C1 — C33
 Jennings C1 — C33

Generators and relations for C33
G = < a,b,c | a3=b3=c3=1, ab=ba, ac=ca, bc=cb >

Character table of C33

 class 1 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 3R 3S 3T 3U 3V 3W 3X 3Y 3Z size 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 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 ζ32 1 1 1 1 1 1 1 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 1 linear of order 3 ρ3 1 ζ3 1 1 1 1 1 1 1 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 1 linear of order 3 ρ4 1 ζ32 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 1 linear of order 3 ρ5 1 ζ3 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 1 1 ζ32 ζ32 ζ32 1 1 1 ζ3 ζ3 1 linear of order 3 ρ6 1 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 1 1 1 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 1 1 1 linear of order 3 ρ7 1 ζ3 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 1 linear of order 3 ρ8 1 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ3 ζ3 ζ3 1 1 1 ζ32 ζ32 ζ32 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 1 1 linear of order 3 ρ9 1 ζ32 1 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ3 1 1 1 ζ3 ζ3 ζ3 1 1 1 ζ32 ζ32 1 linear of order 3 ρ10 1 ζ32 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ3 linear of order 3 ρ11 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 linear of order 3 ρ12 1 1 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 linear of order 3 ρ13 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 linear of order 3 ρ14 1 1 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 1 ζ3 ζ32 ζ32 1 ζ3 1 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ15 1 ζ32 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 ζ32 1 ζ3 1 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ32 1 ζ3 1 ζ3 ζ3 linear of order 3 ρ16 1 1 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 ζ3 linear of order 3 ρ17 1 ζ32 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 1 ζ3 ζ32 1 ζ3 ζ3 ζ32 1 1 ζ3 ζ3 linear of order 3 ρ18 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 ζ32 1 ζ3 ζ3 ζ32 1 1 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 1 ζ3 linear of order 3 ρ19 1 ζ3 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ32 linear of order 3 ρ20 1 1 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 linear of order 3 ρ21 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ3 1 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 linear of order 3 ρ22 1 1 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 ζ32 linear of order 3 ρ23 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ3 1 1 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 1 ζ32 linear of order 3 ρ24 1 ζ3 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 1 ζ32 ζ3 1 ζ32 ζ32 ζ3 1 1 ζ32 ζ32 linear of order 3 ρ25 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 linear of order 3 ρ26 1 ζ3 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 ζ3 1 ζ32 1 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ3 1 ζ32 1 ζ32 ζ32 linear of order 3 ρ27 1 1 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 ζ32 ζ3 1 ζ3 1 ζ32 1 ζ32 ζ3 ζ3 1 ζ32 1 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3

Permutation representations of C33
Regular action on 27 points - transitive group 27T4
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)```

`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)>;`

`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) );`

`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)]])`

`G:=TransitiveGroup(27,4);`

C33 is a maximal subgroup of   C33⋊C2  C32⋊C9  C3≀C3  C33⋊C13
C33 is a maximal quotient of   C9○He3

Matrix representation of C33 in GL3(𝔽7) generated by

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

C33 in GAP, Magma, Sage, TeX

`C_3^3`
`% in TeX`

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

`G:=SmallGroup(27,5);`
`// by ID`

`G=gap.SmallGroup(27,5);`
`# by ID`

`G:=PCGroup([3,-3,3,3]);`
`// Polycyclic`

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

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