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G = C33  order 27 = 33

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

direct product, p-group, elementary abelian, monomial

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 >

C1
C3
C3
C3
C3
C3
C3
C3
C3
C3
C3
C3
C3
C3
C32
C32
C32
C32
C32
C32
C32
C32
C32
C32
C32
C32
C32
C33

Character table of C33

 class 13A3B3C3D3E3F3G3H3I3J3K3L3M3N3O3P3Q3R3S3T3U3V3W3X3Y3Z
 size 111111111111111111111111111
ρ1111111111111111111111111111    trivial
ρ21ζ321111111ζ3ζ3ζ3ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ32ζ32ζ321    linear of order 3
ρ31ζ31111111ζ32ζ32ζ32ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ3ζ3ζ31    linear of order 3
ρ41ζ321ζ3ζ3ζ3ζ32ζ32ζ32111ζ3ζ3ζ3ζ32ζ32ζ32111ζ3ζ3ζ3ζ32ζ321    linear of order 3
ρ51ζ31ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ32111ζ32ζ32ζ32111ζ3ζ31    linear of order 3
ρ6111ζ3ζ3ζ3ζ32ζ32ζ32ζ32ζ32ζ32111ζ3ζ3ζ3ζ3ζ3ζ3ζ32ζ32ζ32111    linear of order 3
ρ71ζ31ζ32ζ32ζ32ζ3ζ3ζ3111ζ32ζ32ζ32ζ3ζ3ζ3111ζ32ζ32ζ32ζ3ζ31    linear of order 3
ρ8111ζ32ζ32ζ32ζ3ζ3ζ3ζ3ζ3ζ3111ζ32ζ32ζ32ζ32ζ32ζ32ζ3ζ3ζ3111    linear of order 3
ρ91ζ321ζ32ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ3111ζ3ζ3ζ3111ζ32ζ321    linear of order 3
ρ101ζ32ζ321ζ3ζ321ζ3ζ321ζ3ζ321ζ3ζ321ζ3ζ321ζ3ζ321ζ3ζ321ζ3ζ3    linear of order 3
ρ111ζ3ζ321ζ3ζ321ζ3ζ32ζ3ζ321ζ3ζ321ζ3ζ321ζ321ζ3ζ321ζ3ζ321ζ3    linear of order 3
ρ1211ζ321ζ3ζ321ζ3ζ32ζ321ζ3ζ321ζ3ζ321ζ3ζ3ζ321ζ3ζ321ζ3ζ32ζ3    linear of order 3
ρ131ζ3ζ32ζ3ζ321ζ321ζ31ζ3ζ32ζ3ζ321ζ321ζ31ζ3ζ32ζ3ζ321ζ321ζ3    linear of order 3
ρ1411ζ32ζ3ζ321ζ321ζ3ζ3ζ321ζ321ζ31ζ3ζ32ζ321ζ31ζ3ζ32ζ3ζ32ζ3    linear of order 3
ρ151ζ32ζ32ζ3ζ321ζ321ζ3ζ321ζ31ζ3ζ32ζ3ζ321ζ3ζ321ζ321ζ31ζ3ζ3    linear of order 3
ρ1611ζ32ζ321ζ3ζ3ζ3211ζ3ζ32ζ321ζ3ζ3ζ3211ζ3ζ32ζ321ζ3ζ3ζ32ζ3    linear of order 3
ρ171ζ32ζ32ζ321ζ3ζ3ζ321ζ3ζ3211ζ3ζ32ζ321ζ3ζ321ζ3ζ3ζ3211ζ3ζ3    linear of order 3
ρ181ζ3ζ32ζ321ζ3ζ3ζ321ζ321ζ3ζ3ζ3211ζ3ζ32ζ3ζ3211ζ3ζ32ζ321ζ3    linear of order 3
ρ191ζ3ζ31ζ32ζ31ζ32ζ31ζ32ζ31ζ32ζ31ζ32ζ31ζ32ζ31ζ32ζ31ζ32ζ32    linear of order 3
ρ2011ζ31ζ32ζ31ζ32ζ3ζ31ζ32ζ31ζ32ζ31ζ32ζ32ζ31ζ32ζ31ζ32ζ3ζ32    linear of order 3
ρ211ζ32ζ31ζ32ζ31ζ32ζ3ζ32ζ31ζ32ζ31ζ32ζ31ζ31ζ32ζ31ζ32ζ31ζ32    linear of order 3
ρ2211ζ3ζ31ζ32ζ32ζ311ζ32ζ3ζ31ζ32ζ32ζ311ζ32ζ3ζ31ζ32ζ32ζ3ζ32    linear of order 3
ρ231ζ32ζ3ζ31ζ32ζ32ζ31ζ31ζ32ζ32ζ311ζ32ζ3ζ32ζ311ζ32ζ3ζ31ζ32    linear of order 3
ρ241ζ3ζ3ζ31ζ32ζ32ζ31ζ32ζ311ζ32ζ3ζ31ζ32ζ31ζ32ζ32ζ311ζ32ζ32    linear of order 3
ρ251ζ32ζ3ζ32ζ31ζ31ζ321ζ32ζ3ζ32ζ31ζ31ζ321ζ32ζ3ζ32ζ31ζ31ζ32    linear of order 3
ρ261ζ3ζ3ζ32ζ31ζ31ζ32ζ31ζ321ζ32ζ3ζ32ζ31ζ32ζ31ζ31ζ321ζ32ζ32    linear of order 3
ρ2711ζ3ζ32ζ31ζ31ζ32ζ32ζ31ζ31ζ321ζ32ζ3ζ31ζ321ζ32ζ3ζ32ζ3ζ32    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

400
010
002
,
100
040
004
,
200
040
001
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
 

Export

Subgroup lattice of C33 in TeX
Character table of C33 in TeX

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