non-abelian, supersoluble, monomial
Aliases: C92.2S3, C92.C3⋊C2, C3.He3.1C6, C32.2(C32⋊C6), C3.6(He3.2C6), 3- 1+2.S3.1C3, (C3×C9).18(C3×S3), SmallGroup(486,38)
Series: Derived ►Chief ►Lower central ►Upper central
| C3.He3 — C92.S3 |
Generators and relations for C92.S3
G = < a,b,c,d | a9=b9=d2=1, c3=b6, ab=ba, cac-1=ab-1, ad=da, cbc-1=a3b7, dbd=a3b-1, dcd=b3c2 >
Permutation representations of C92.S3
►On 27 points - transitive group 27T184
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 14 23 3 17 20 2 11 26)(4 15 22 6 18 19 5 12 25)(7 16 21 9 10 27 8 13 24)
(2 3)(4 8)(5 7)(6 9)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 19)(17 20)(18 21)
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), (1,14,23,3,17,20,2,11,26)(4,15,22,6,18,19,5,12,25)(7,16,21,9,10,27,8,13,24), (2,3)(4,8)(5,7)(6,9)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21)>;
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), (1,14,23,3,17,20,2,11,26)(4,15,22,6,18,19,5,12,25)(7,16,21,9,10,27,8,13,24), (2,3)(4,8)(5,7)(6,9)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,19)(17,20)(18,21) );
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)], [(1,14,23,3,17,20,2,11,26),(4,15,22,6,18,19,5,12,25),(7,16,21,9,10,27,8,13,24)], [(2,3),(4,8),(5,7),(6,9),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,19),(17,20),(18,21)]])
G:=TransitiveGroup(27,184);
31 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 6A | 6B | 9A | ··· | 9F | 9G | ··· | 9O | 9P | 9Q | 9R | 18A | ··· | 18F |
| order | 1 | 2 | 3 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 9 | ··· | 9 | 9 | 9 | 9 | 18 | ··· | 18 |
| size | 1 | 27 | 2 | 3 | 3 | 27 | 27 | 3 | ··· | 3 | 6 | ··· | 6 | 54 | 54 | 54 | 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.C3 | 3- 1+2.S3 | C3.He3 | 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 |
| 9 | 9 | 7 | 5 | 0 | 0 |
| 14 | 14 | 14 | 2 | 0 | 0 |
| 14 | 14 | 0 | 0 | 2 | 14 |
| 9 | 9 | 0 | 0 | 5 | 7 |
| 7 | 14 | 0 | 0 | 0 | 0 |
| 5 | 2 | 0 | 0 | 0 | 0 |
| 0 | 14 | 5 | 7 | 0 | 0 |
| 7 | 2 | 12 | 17 | 0 | 0 |
| 5 | 0 | 0 | 0 | 2 | 14 |
| 0 | 14 | 0 | 0 | 5 | 7 |
| 1 | 1 | 0 | 0 | 18 | 17 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 | 0 | 18 |
| 0 | 0 | 1 | 0 | 0 | 18 |
| 0 | 0 | 0 | 1 | 0 | 18 |
| 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))| [1,0,9,14,14,9,0,1,9,14,14,9,0,0,7,14,0,0,0,0,5,2,0,0,0,0,0,0,2,5,0,0,0,0,14,7],[7,5,0,7,5,0,14,2,14,2,0,14,0,0,5,12,0,0,0,0,7,17,0,0,0,0,0,0,2,5,0,0,0,0,14,7],[1,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,1,0,0,0,0,17,18,18,18,18,18],[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.S_3
% in TeX
G:=Group("C9^2.S3"); // GroupNames label
G:=SmallGroup(486,38);
// by ID
G=gap.SmallGroup(486,38);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,331,3134,224,986,6051,951,453,1096,11669]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^9=d^2=1,c^3=b^6,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=b^3*c^2>;
// generators/relations
Export