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G = C32⋊C9order 81 = 34

The semidirect product of C32 and C9 acting via C9/C3=C3

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

Aliases: C32⋊C9, C3.1He3, C33.1C3, C32.6C32, C3.13- 1+2, (C3×C9)⋊1C3, C3.1(C3×C9), SmallGroup(81,3)

Series: Derived Chief Lower central Upper central Jennings

C1C3 — C32⋊C9
C1C3C32C33 — C32⋊C9
C1C3 — C32⋊C9
C1C32 — C32⋊C9
C1C3C32 — C32⋊C9

Generators and relations for C32⋊C9
 G = < a,b,c | a3=b3=c9=1, ab=ba, cac-1=ab-1, bc=cb >

3C3
3C3
3C3
3C9
3C32
3C9
3C9
3C32
3C32

Permutation representations of C32⋊C9
On 27 points - transitive group 27T17
Generators in S27
(2 15 21)(3 22 16)(5 18 24)(6 25 10)(8 12 27)(9 19 13)
(1 20 14)(2 21 15)(3 22 16)(4 23 17)(5 24 18)(6 25 10)(7 26 11)(8 27 12)(9 19 13)
(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)

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

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

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

G:=TransitiveGroup(27,17);

C32⋊C9 is a maximal subgroup of
C32⋊C18  C32⋊D9  C322D9  C32.24He3  C33.C32  C33.3C32  C32.27He3  C32.28He3  C32.29He3  C33.7C32  C33⋊C9  C32.19He3  C32.20He3  He3⋊C9  3- 1+2⋊C9  C923C3  C9×He3  C9×3- 1+2  C34.C3  C9⋊He3  C32.23C33  C9⋊3- 1+2  C33.31C32  C927C3  C924C3  C925C3  C928C3  C62.16C32  C62⋊C9
C32⋊C9 is a maximal quotient of
C3.C92  C32⋊C27  C33⋊C9  C32.19He3  C32.20He3  C9.4He3  He3⋊C9  3- 1+2⋊C9  C9.5He3  C9.6He3  C62.16C32  C62⋊C9

33 conjugacy classes

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

33 irreducible representations

dim111133
type+
imageC1C3C3C9He33- 1+2
kernelC32⋊C9C3×C9C33C32C3C3
# reps1621824

Matrix representation of C32⋊C9 in GL4(𝔽19) generated by

7000
0700
00110
0001
,
1000
0700
0070
0007
,
17000
0010
0001
0700
G:=sub<GL(4,GF(19))| [7,0,0,0,0,7,0,0,0,0,11,0,0,0,0,1],[1,0,0,0,0,7,0,0,0,0,7,0,0,0,0,7],[17,0,0,0,0,0,0,7,0,1,0,0,0,0,1,0] >;

C32⋊C9 in GAP, Magma, Sage, TeX

C_3^2\rtimes C_9
% in TeX

G:=Group("C3^2:C9");
// GroupNames label

G:=SmallGroup(81,3);
// by ID

G=gap.SmallGroup(81,3);
# by ID

G:=PCGroup([4,-3,3,-3,3,108,97]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^9=1,a*b=b*a,c*a*c^-1=a*b^-1,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of C32⋊C9 in TeX

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