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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

C1C3 — C3×3- 1+2
C1C3C32C33 — C3×3- 1+2
C1C3 — C3×3- 1+2
C1C32 — C3×3- 1+2
C1C3C3 — 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 >

3C3
3C3
3C3
3C32
3C32
3C32

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

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

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

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

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···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

Export

Subgroup lattice of C3×3- 1+2 in TeX

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