p-group, metabelian, nilpotent (class 2), monomial
Aliases: C32:C9, C3.1He3, C33.1C3, C32.6C32, C3.13- 1+2, (C3xC9):1C3, C3.1(C3xC9), SmallGroup(81,3)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C32:C9
G = < a,b,c | a3=b3=c9=1, ab=ba, cac-1=ab-1, bc=cb >
Quotients: C1, C3, C9, C32, C3xC9, He3, 3- 1+2, C32:C9
Permutation representations of C32:C9
►On 27 points - transitive group 27T17
(2 11 21)(3 22 12)(5 14 24)(6 25 15)(8 17 27)(9 19 18)
(1 20 10)(2 21 11)(3 22 12)(4 23 13)(5 24 14)(6 25 15)(7 26 16)(8 27 17)(9 19 18)
(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)| (2,11,21)(3,22,12)(5,14,24)(6,25,15)(8,17,27)(9,19,18), (1,20,10)(2,21,11)(3,22,12)(4,23,13)(5,24,14)(6,25,15)(7,26,16)(8,27,17)(9,19,18), (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( (2,11,21)(3,22,12)(5,14,24)(6,25,15)(8,17,27)(9,19,18), (1,20,10)(2,21,11)(3,22,12)(4,23,13)(5,24,14)(6,25,15)(7,26,16)(8,27,17)(9,19,18), (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([[(2,11,21),(3,22,12),(5,14,24),(6,25,15),(8,17,27),(9,19,18)], [(1,20,10),(2,21,11),(3,22,12),(4,23,13),(5,24,14),(6,25,15),(7,26,16),(8,27,17),(9,19,18)], [(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 C32:2D9 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 C92:3C3 C9xHe3 C9x3- 1+2 C34.C3 C9:He3 C32.23C33 C9:3- 1+2 C33.31C32 C92:7C3 C92:4C3 C92:5C3 C92:8C3 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 | ··· | 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 | 3 |
| type | + | |||||
| image | C1 | C3 | C3 | C9 | He3 | 3- 1+2 |
| kernel | C32:C9 | C3xC9 | C33 | C32 | C3 | C3 |
| # reps | 1 | 6 | 2 | 18 | 2 | 4 |
Matrix representation of C32:C9 ►in GL4(F19) generated by
| 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 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 7 | 0 | 0 |
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
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