p-group, metabelian, nilpotent (class 4), monomial
Aliases: C92.2C3, C32.3He3, (C3×C9).18C32, C3.He3.1C3, C3.8(He3⋊C3), SmallGroup(243,27)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C92.C3
G = < a,b,c | a9=b9=1, c3=b6, ab=ba, cac-1=a7b-1, cbc-1=a3b >
Permutation representations of C92.C3
►On 27 points - transitive group 27T112
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 11 12 13 14 15 16 17 18)(19 21 23 25 27 20 22 24 26)
(1 26 16 2 20 13 3 23 10)(4 21 15 5 24 12 6 27 18)(7 19 11 8 22 17 9 25 14)
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)(10,11,12,13,14,15,16,17,18)(19,21,23,25,27,20,22,24,26), (1,26,16,2,20,13,3,23,10)(4,21,15,5,24,12,6,27,18)(7,19,11,8,22,17,9,25,14)>;
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)(10,11,12,13,14,15,16,17,18)(19,21,23,25,27,20,22,24,26), (1,26,16,2,20,13,3,23,10)(4,21,15,5,24,12,6,27,18)(7,19,11,8,22,17,9,25,14) );
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),(10,11,12,13,14,15,16,17,18),(19,21,23,25,27,20,22,24,26)], [(1,26,16,2,20,13,3,23,10),(4,21,15,5,24,12,6,27,18),(7,19,11,8,22,17,9,25,14)]])
G:=TransitiveGroup(27,112);
C92.C3 is a maximal subgroup of
C92.S3
35 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 9A | ··· | 9X | 9Y | ··· | 9AD |
| order | 1 | 3 | 3 | 3 | 3 | 9 | ··· | 9 | 9 | ··· | 9 |
| size | 1 | 1 | 1 | 3 | 3 | 3 | ··· | 3 | 27 | ··· | 27 |
35 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C3 | He3 | He3⋊C3 | C92.C3 |
| kernel | C92.C3 | C92 | C3.He3 | C32 | C3 | C1 |
| # reps | 1 | 2 | 6 | 2 | 6 | 18 |
Matrix representation of C92.C3 ►in GL3(𝔽19) generated by
| 7 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 17 |
| 6 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 6 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 11 | 0 | 0 |
G:=sub<GL(3,GF(19))| [7,0,0,0,4,0,0,0,17],[6,0,0,0,9,0,0,0,6],[0,0,11,1,0,0,0,1,0] >;
C92.C3 in GAP, Magma, Sage, TeX
C_9^2.C_3
% in TeX
G:=Group("C9^2.C3"); // GroupNames label
G:=SmallGroup(243,27);
// by ID
G=gap.SmallGroup(243,27);
# by ID
G:=PCGroup([5,-3,3,-3,-3,-3,405,121,456,542,282,2163]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=1,c^3=b^6,a*b=b*a,c*a*c^-1=a^7*b^-1,c*b*c^-1=a^3*b>;
// generators/relations
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