non-abelian, supersoluble, monomial
Aliases: C33⋊1D9, C34.1S3, C3.4C3≀S3, C32⋊C9⋊3C6, C33⋊C9⋊2C2, C32.5(C3×D9), C32⋊2D9⋊4C3, C32.2(C9⋊C6), C33.26(C3×S3), C3.6(C32⋊D9), C32.35(C32⋊C6), SmallGroup(486,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C9 — C33⋊1D9 |
Generators and relations for C33⋊1D9
G = < a,b,c,d,e | a3=b3=c3=d9=e2=1, ab=ba, ac=ca, dad-1=eae=abc-1, bc=cb, dbd-1=bc-1, ebe=b-1c-1, cd=dc, ce=ec, ede=d-1 >
Subgroups: 668 in 105 conjugacy classes, 14 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C6, C3×C9, C33, C33, C3×D9, S3×C32, C3×C3⋊S3, C32⋊C9, C32⋊C9, C34, C32⋊2D9, C32×C3⋊S3, C33⋊C9, C33⋊1D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, C3×D9, C32⋊C6, C9⋊C6, C32⋊D9, C3≀S3, C33⋊1D9
►On 18 points - transitive group 18T172
Generators in S18
(2 8 5)(12 15 18)
(2 8 5)(3 6 9)(10 16 13)(12 15 18)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)
G:=sub<Sym(18)| (2,8,5)(12,15,18), (2,8,5)(3,6,9)(10,16,13)(12,15,18), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)>;
G:=Group( (2,8,5)(12,15,18), (2,8,5)(3,6,9)(10,16,13)(12,15,18), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10) );
G=PermutationGroup([[(2,8,5),(12,15,18)], [(2,8,5),(3,6,9),(10,16,13),(12,15,18)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]])
G:=TransitiveGroup(18,172);
►On 27 points - transitive group 27T188
Generators in S27
(2 22 15)(3 26 13)(5 25 18)(6 20 16)(8 19 12)(9 23 10)
(1 11 24)(2 22 15)(3 6 9)(4 14 27)(5 25 18)(7 17 21)(8 19 12)(10 13 16)(20 23 26)
(1 27 17)(2 19 18)(3 20 10)(4 21 11)(5 22 12)(6 23 13)(7 24 14)(8 25 15)(9 26 16)
(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 9)(2 8)(3 7)(4 6)(10 14)(11 13)(15 18)(16 17)(19 25)(20 24)(21 23)(26 27)
G:=sub<Sym(27)| (2,22,15)(3,26,13)(5,25,18)(6,20,16)(8,19,12)(9,23,10), (1,11,24)(2,22,15)(3,6,9)(4,14,27)(5,25,18)(7,17,21)(8,19,12)(10,13,16)(20,23,26), (1,27,17)(2,19,18)(3,20,10)(4,21,11)(5,22,12)(6,23,13)(7,24,14)(8,25,15)(9,26,16), (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), (1,9)(2,8)(3,7)(4,6)(10,14)(11,13)(15,18)(16,17)(19,25)(20,24)(21,23)(26,27)>;
G:=Group( (2,22,15)(3,26,13)(5,25,18)(6,20,16)(8,19,12)(9,23,10), (1,11,24)(2,22,15)(3,6,9)(4,14,27)(5,25,18)(7,17,21)(8,19,12)(10,13,16)(20,23,26), (1,27,17)(2,19,18)(3,20,10)(4,21,11)(5,22,12)(6,23,13)(7,24,14)(8,25,15)(9,26,16), (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), (1,9)(2,8)(3,7)(4,6)(10,14)(11,13)(15,18)(16,17)(19,25)(20,24)(21,23)(26,27) );
G=PermutationGroup([[(2,22,15),(3,26,13),(5,25,18),(6,20,16),(8,19,12),(9,23,10)], [(1,11,24),(2,22,15),(3,6,9),(4,14,27),(5,25,18),(7,17,21),(8,19,12),(10,13,16),(20,23,26)], [(1,27,17),(2,19,18),(3,20,10),(4,21,11),(5,22,12),(6,23,13),(7,24,14),(8,25,15),(9,26,16)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,14),(11,13),(15,18),(16,17),(19,25),(20,24),(21,23),(26,27)]])
G:=TransitiveGroup(27,188);
►On 27 points - transitive group 27T189
Generators in S27
(2 19 18)(3 20 10)(5 22 12)(6 23 13)(8 25 15)(9 26 16)
(1 17 27)(2 19 18)(4 11 21)(5 22 12)(7 14 24)(8 25 15)
(1 27 17)(2 19 18)(3 20 10)(4 21 11)(5 22 12)(6 23 13)(7 24 14)(8 25 15)(9 26 16)
(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 9)(2 8)(3 7)(4 6)(10 14)(11 13)(15 18)(16 17)(19 25)(20 24)(21 23)(26 27)
G:=sub<Sym(27)| (2,19,18)(3,20,10)(5,22,12)(6,23,13)(8,25,15)(9,26,16), (1,17,27)(2,19,18)(4,11,21)(5,22,12)(7,14,24)(8,25,15), (1,27,17)(2,19,18)(3,20,10)(4,21,11)(5,22,12)(6,23,13)(7,24,14)(8,25,15)(9,26,16), (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), (1,9)(2,8)(3,7)(4,6)(10,14)(11,13)(15,18)(16,17)(19,25)(20,24)(21,23)(26,27)>;
G:=Group( (2,19,18)(3,20,10)(5,22,12)(6,23,13)(8,25,15)(9,26,16), (1,17,27)(2,19,18)(4,11,21)(5,22,12)(7,14,24)(8,25,15), (1,27,17)(2,19,18)(3,20,10)(4,21,11)(5,22,12)(6,23,13)(7,24,14)(8,25,15)(9,26,16), (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), (1,9)(2,8)(3,7)(4,6)(10,14)(11,13)(15,18)(16,17)(19,25)(20,24)(21,23)(26,27) );
G=PermutationGroup([[(2,19,18),(3,20,10),(5,22,12),(6,23,13),(8,25,15),(9,26,16)], [(1,17,27),(2,19,18),(4,11,21),(5,22,12),(7,14,24),(8,25,15)], [(1,27,17),(2,19,18),(3,20,10),(4,21,11),(5,22,12),(6,23,13),(7,24,14),(8,25,15),(9,26,16)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,14),(11,13),(15,18),(16,17),(19,25),(20,24),(21,23),(26,27)]])
G:=TransitiveGroup(27,189);
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | ··· | 3K | 3L | ··· | 3T | 6A | ··· | 6H | 9A | ··· | 9I |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 27 | ··· | 27 | 18 | ··· | 18 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | D9 | C3×S3 | C3×D9 | C3≀S3 | C32⋊C6 | C9⋊C6 | C33⋊1D9 |
| kernel | C33⋊1D9 | C33⋊C9 | C32⋊2D9 | C32⋊C9 | C34 | C33 | C33 | C32 | C3 | C32 | C32 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 3 | 2 | 6 | 12 | 1 | 2 | 6 |
Matrix representation of C33⋊1D9 ►in GL5(𝔽19)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 12 | 7 | 0 |
| 0 | 0 | 12 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 7 | 11 | 0 |
| 0 | 0 | 18 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 11 | 1 | 0 | 0 | 0 |
| 12 | 15 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 6 |
| 0 | 0 | 0 | 0 | 18 |
| 0 | 0 | 0 | 1 | 18 |
| 0 | 12 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 6 |
| 0 | 0 | 0 | 1 | 18 |
| 0 | 0 | 0 | 0 | 18 |
G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,11,12,12,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,7,7,18,0,0,0,11,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11],[11,12,0,0,0,1,15,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,6,18,18],[0,8,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,6,18,18] >;
C33⋊1D9 in GAP, Magma, Sage, TeX
C_3^3\rtimes_1D_9
% in TeX
G:=Group("C3^3:1D9"); // GroupNames label
G:=SmallGroup(486,19);
// by ID
G=gap.SmallGroup(486,19);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,224,338,4755,735,3244]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^9=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a*b*c^-1,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1*c^-1,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations