p-group, metabelian, nilpotent (class 4), monomial
Aliases: C92⋊1C3, C32.1He3, C3.He3⋊1C3, (C3×C9).16C32, He3⋊C3.1C3, C3.6(He3⋊C3), SmallGroup(243,25)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C92⋊C3
G = < a,b,c | a9=b9=c3=1, ab=ba, cac-1=a7b2, cbc-1=a3b7 >
Permutation representations of C92⋊C3
►On 27 points - transitive group 27T108
Generators in S27
(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 7 4 3 9 6 2 8 5)(10 14 18 13 17 12 16 11 15)(19 21 23 25 27 20 22 24 26)
(1 23 10)(2 26 16)(3 20 13)(4 21 18)(5 24 15)(6 27 12)(7 22 14)(8 25 11)(9 19 17)
G:=sub<Sym(27)| (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), (1,7,4,3,9,6,2,8,5)(10,14,18,13,17,12,16,11,15)(19,21,23,25,27,20,22,24,26), (1,23,10)(2,26,16)(3,20,13)(4,21,18)(5,24,15)(6,27,12)(7,22,14)(8,25,11)(9,19,17)>;
G:=Group( (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), (1,7,4,3,9,6,2,8,5)(10,14,18,13,17,12,16,11,15)(19,21,23,25,27,20,22,24,26), (1,23,10)(2,26,16)(3,20,13)(4,21,18)(5,24,15)(6,27,12)(7,22,14)(8,25,11)(9,19,17) );
G=PermutationGroup([[(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)], [(1,7,4,3,9,6,2,8,5),(10,14,18,13,17,12,16,11,15),(19,21,23,25,27,20,22,24,26)], [(1,23,10),(2,26,16),(3,20,13),(4,21,18),(5,24,15),(6,27,12),(7,22,14),(8,25,11),(9,19,17)]])
G:=TransitiveGroup(27,108);
C92⋊C3 is a maximal subgroup of
C92⋊C6
35 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 3E | 3F | 9A | ··· | 9X | 9Y | 9Z | 9AA | 9AB |
| order | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 9 | 9 | 9 | 9 |
| size | 1 | 1 | 1 | 3 | 3 | 27 | 27 | 3 | ··· | 3 | 27 | 27 | 27 | 27 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | C3 | He3 | He3⋊C3 | C92⋊C3 |
| kernel | C92⋊C3 | C92 | He3⋊C3 | C3.He3 | C32 | C3 | C1 |
| # reps | 1 | 2 | 2 | 4 | 2 | 6 | 18 |
Matrix representation of C92⋊C3 ►in GL3(𝔽19) generated by
| 7 | 0 | 0 |
| 13 | 4 | 0 |
| 15 | 0 | 5 |
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 6 | 0 | 9 |
| 1 | 10 | 0 |
| 0 | 18 | 1 |
| 0 | 18 | 0 |
G:=sub<GL(3,GF(19))| [7,13,15,0,4,0,0,0,5],[6,0,6,0,6,0,0,0,9],[1,0,0,10,18,18,0,1,0] >;
C92⋊C3 in GAP, Magma, Sage, TeX
C_9^2\rtimes C_3
% in TeX
G:=Group("C9^2:C3"); // GroupNames label
G:=SmallGroup(243,25);
// by ID
G=gap.SmallGroup(243,25);
# by ID
G:=PCGroup([5,-3,3,-3,-3,-3,121,186,542,282,2163]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^3=1,a*b=b*a,c*a*c^-1=a^7*b^2,c*b*c^-1=a^3*b^7>;
// generators/relations
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