non-abelian, supersoluble, monomial
Aliases: C92⋊1S3, C92⋊2C3⋊1C2, He3⋊S3⋊1C3, He3⋊C3.1C6, C32.1(C32⋊C6), C3.5(He3.2C6), (C3×C9).17(C3×S3), SmallGroup(486,36)
Series: Derived ►Chief ►Lower central ►Upper central
| He3⋊C3 — C92⋊S3 |
Generators and relations for C92⋊S3
G = < a,b,c,d | a9=b9=c3=d2=1, ab=ba, cac-1=ab-1, ad=da, cbc-1=a3b7, dbd=a3b-1, dcd=c-1 >
Permutation representations of C92⋊S3
►On 27 points - transitive group 27T185
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 12 14 16 18 11 13 15 17)(19 20 21 22 23 24 25 26 27)
(1 10 21)(2 16 24)(3 13 27)(4 11 20)(5 17 23)(6 14 26)(7 12 19)(8 18 22)(9 15 25)
(2 3)(4 8)(5 7)(6 9)(10 21)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 19)(18 20)
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)(10,12,14,16,18,11,13,15,17)(19,20,21,22,23,24,25,26,27), (1,10,21)(2,16,24)(3,13,27)(4,11,20)(5,17,23)(6,14,26)(7,12,19)(8,18,22)(9,15,25), (2,3)(4,8)(5,7)(6,9)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,19)(18,20)>;
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)(10,12,14,16,18,11,13,15,17)(19,20,21,22,23,24,25,26,27), (1,10,21)(2,16,24)(3,13,27)(4,11,20)(5,17,23)(6,14,26)(7,12,19)(8,18,22)(9,15,25), (2,3)(4,8)(5,7)(6,9)(10,21)(11,22)(12,23)(13,24)(14,25)(15,26)(16,27)(17,19)(18,20) );
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),(10,12,14,16,18,11,13,15,17),(19,20,21,22,23,24,25,26,27)], [(1,10,21),(2,16,24),(3,13,27),(4,11,20),(5,17,23),(6,14,26),(7,12,19),(8,18,22),(9,15,25)], [(2,3),(4,8),(5,7),(6,9),(10,21),(11,22),(12,23),(13,24),(14,25),(15,26),(16,27),(17,19),(18,20)]])
G:=TransitiveGroup(27,185);
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9O | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 18 | ··· | 18 |
| size | 1 | 27 | 2 | 3 | 3 | 54 | 54 | 54 | 27 | 27 | 3 | ··· | 3 | 6 | ··· | 6 | 27 | ··· | 27 |
31 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 6 | 6 | 6 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C3 | C6 | S3 | C3×S3 | He3.2C6 | C32⋊C6 | C92⋊S3 | C92⋊S3 |
| kernel | C92⋊S3 | C92⋊2C3 | He3⋊S3 | He3⋊C3 | C92 | C3×C9 | C3 | C32 | C1 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 12 | 1 | 3 | 6 |
Matrix representation of C92⋊S3 ►in GL6(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 5 | 0 | 0 |
| 0 | 0 | 14 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 7 | 0 | 0 |
| 0 | 0 | 12 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 |
| 0 | 0 | 0 | 0 | 5 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,14,0,0,0,0,5,7,0,0,0,0,0,0,7,5,0,0,0,0,14,2],[7,5,0,0,0,0,14,2,0,0,0,0,0,0,17,12,0,0,0,0,7,5,0,0,0,0,0,0,7,5,0,0,0,0,14,2],[0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
C92⋊S3 in GAP, Magma, Sage, TeX
C_9^2\rtimes S_3
% in TeX
G:=Group("C9^2:S3"); // GroupNames label
G:=SmallGroup(486,36);
// by ID
G=gap.SmallGroup(486,36);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,218,224,6051,951,453,1096,11669]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^9=c^3=d^2=1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,c*b*c^-1=a^3*b^7,d*b*d=a^3*b^-1,d*c*d=c^-1>;
// generators/relations
Export