p-group, metabelian, nilpotent (class 4), monomial
Aliases: C92⋊2C3, C32.2He3, He3⋊C3⋊1C3, (C3×C9).17C32, C3.7(He3⋊C3), 3-Sylow(SL(3,19)), SmallGroup(243,26)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C92⋊2C3
G = < a,b,c | a9=b9=c3=1, ab=ba, cac-1=a7b-1, cbc-1=a3b >
Permutation representations of C92⋊2C3
►On 27 points - transitive group 27T104
Generators in S27
(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 8 5 2 9 6 3 7 4)(10 17 15 13 11 18 16 14 12)(19 27 26 25 24 23 22 21 20)
(1 27 10)(2 24 13)(3 21 16)(4 19 18)(5 25 12)(6 22 15)(7 20 17)(8 26 11)(9 23 14)
G:=sub<Sym(27)| (10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,8,5,2,9,6,3,7,4)(10,17,15,13,11,18,16,14,12)(19,27,26,25,24,23,22,21,20), (1,27,10)(2,24,13)(3,21,16)(4,19,18)(5,25,12)(6,22,15)(7,20,17)(8,26,11)(9,23,14)>;
G:=Group( (10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,8,5,2,9,6,3,7,4)(10,17,15,13,11,18,16,14,12)(19,27,26,25,24,23,22,21,20), (1,27,10)(2,24,13)(3,21,16)(4,19,18)(5,25,12)(6,22,15)(7,20,17)(8,26,11)(9,23,14) );
G=PermutationGroup([[(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(1,8,5,2,9,6,3,7,4),(10,17,15,13,11,18,16,14,12),(19,27,26,25,24,23,22,21,20)], [(1,27,10),(2,24,13),(3,21,16),(4,19,18),(5,25,12),(6,22,15),(7,20,17),(8,26,11),(9,23,14)]])
G:=TransitiveGroup(27,104);
C92⋊2C3 is a maximal subgroup of
C92⋊S3 C92⋊2C6 C92⋊2S3
35 conjugacy classes
| class | 1 | 3A | 3B | 3C | 3D | 3E | ··· | 3J | 9A | ··· | 9X |
| order | 1 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | 1 | 3 | 3 | 27 | ··· | 27 | 3 | ··· | 3 |
35 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C3 | He3 | He3⋊C3 | C92⋊2C3 |
| kernel | C92⋊2C3 | C92 | He3⋊C3 | C32 | C3 | C1 |
| # reps | 1 | 2 | 6 | 2 | 6 | 18 |
Matrix representation of C92⋊2C3 ►in GL3(𝔽19) generated by
| 7 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 9 |
| 5 | 0 | 0 |
| 0 | 16 | 0 |
| 0 | 0 | 5 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(19))| [7,0,0,0,16,0,0,0,9],[5,0,0,0,16,0,0,0,5],[0,0,1,1,0,0,0,1,0] >;
C92⋊2C3 in GAP, Magma, Sage, TeX
C_9^2\rtimes_2C_3
% in TeX
G:=Group("C9^2:2C3"); // GroupNames label
G:=SmallGroup(243,26);
// by ID
G=gap.SmallGroup(243,26);
# by ID
G:=PCGroup([5,-3,3,-3,-3,-3,121,456,542,282,2163]);
// Polycyclic
G:=Group<a,b,c|a^9=b^9=c^3=1,a*b=b*a,c*a*c^-1=a^7*b^-1,c*b*c^-1=a^3*b>;
// generators/relations
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