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G = C3×3- 1+2  order 81 = 34

Direct product of C3 and 3- 1+2

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C3×3- 1+2, C9⋊C32, C33.2C3, C3.2C33, C32.8C32, (C3×C9)⋊4C3, SmallGroup(81,13)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C3 — C3×3- 1+2
Chief series
C1 — C3 — C32 — C33 — C3×3- 1+2
Lower central
C1 — C3 — C3×3- 1+2
Upper central
C1 — C32 — C3×3- 1+2
Jennings
C1 — C3 — C3 — C3×3- 1+2

Generators and relations for C3×3- 1+2
 G = < a,b,c | a3=b9=c3=1, ab=ba, ac=ca, cbc-1=b4 >

C1
C3
C3
C3
C3
3C3
3C3
3C3
C9
C32
C9
C9
C32
C9
C9
C9
C9
C32
C9
C9
C32
3C32
3C32
3C32
3- 1+2
3- 1+2
3- 1+2
3- 1+2
3- 1+2
3- 1+2
3- 1+2
3- 1+2
C3×C9
3- 1+2
C3×C9
C3×C9
C33
C3×3- 1+2

Permutation representations of C3×3- 1+2
►On 27 points - transitive group 27T16
Generators in S27
(1 13 21)(2 14 22)(3 15 23)(4 16 24)(5 17 25)(6 18 26)(7 10 27)(8 11 19)(9 12 20)

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

C3×3- 1+2 is a maximal subgroup of
 C33.S3  C33.C32  C33.3C32  C32.28He3  3- 1+2⋊C9  C34.C3  C9⋊He3  C32.23C33  C9⋊3- 1+2  C92⋊7C3  C92⋊8C3  C92⋊9C3  C33⋊C32  He3.C32  He3⋊C32  C32.C33  C9.2He3  3- 1+4
C3×3- 1+2 is a maximal quotient of
 C34.C3  C9⋊He3  C9⋊3- 1+2  C33.31C32  C92⋊7C3  C92⋊8C3  C92⋊9C3

33 conjugacy classes

class 1 3A···3H3I···3N9A···9R
order13···33···39···9
size11···13···33···3

33 irreducible representations

dim11113
type+
imageC1C3C3C33- 1+2
kernelC3×3- 1+2C3×C93- 1+2C33C3
# reps161826

Matrix representation of C3×3- 1+2 ►in GL4(𝔽19) generated by

7000
01100
00110
00011
,
7000
0070
0181810
012121
,
7000
0100
00110
0787
G:=sub<GL(4,GF(19))| [7,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[7,0,0,0,0,0,18,12,0,7,18,12,0,0,10,1],[7,0,0,0,0,1,0,7,0,0,11,8,0,0,0,7] >;
 

C3×3- 1+2 in GAP, Magma, Sage, TeX 

C_3\times 3_-^{1+2}
 
% in TeX
 
G:=Group("C3xES-(3,1)");
 
// GroupNames label
 
G:=SmallGroup(81,13);
 
// by ID
 
G=gap.SmallGroup(81,13);
 
# by ID
 
G:=PCGroup([4,-3,3,3,-3,108,241]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^9=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
 
// generators/relations
 

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Subgroup lattice of C3×3- 1+2 in TeX

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