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## G = C3×3- 1+2order 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 >

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  C927C3  C928C3  C929C3  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  C927C3  C928C3  C929C3

33 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3N 9A ··· 9R order 1 3 ··· 3 3 ··· 3 9 ··· 9 size 1 1 ··· 1 3 ··· 3 3 ··· 3

33 irreducible representations

 dim 1 1 1 1 3 type + image C1 C3 C3 C3 3- 1+2 kernel C3×3- 1+2 C3×C9 3- 1+2 C33 C3 # reps 1 6 18 2 6

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

 7 0 0 0 0 11 0 0 0 0 11 0 0 0 0 11
,
 7 0 0 0 0 0 7 0 0 18 18 10 0 12 12 1
,
 7 0 0 0 0 1 0 0 0 0 11 0 0 7 8 7
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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