p-group, metabelian, nilpotent (class 3), monomial
Aliases: C33⋊1C9, C34.1C3, C32.18He3, C33.18C32, C32.43- 1+2, C3.1C3≀C3, C32⋊C9⋊4C3, C32.7(C3×C9), C3.3(C32⋊C9), SmallGroup(243,13)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C33⋊C9
G = < a,b,c,d | a3=b3=c3=d9=1, ab=ba, ac=ca, dad-1=ab-1c, bc=cb, dbd-1=bc-1, cd=dc >
Subgroups: 252 in 90 conjugacy classes, 18 normal (7 characteristic)
C1, C3, C3, C3, C9, C32, C32, C32, C3×C9, C33, C33, C33, C32⋊C9, C34, C33⋊C9
Quotients: C1, C3, C9, C32, C3×C9, He3, 3- 1+2, C32⋊C9, C3≀C3, C33⋊C9
►On 27 points - transitive group 27T89
Generators in S27
(1 21 14)(2 25 12)(3 16 23)(4 24 17)(5 19 15)(6 10 26)(7 27 11)(8 22 18)(9 13 20)
(1 21 14)(2 8 5)(3 10 20)(4 24 17)(6 13 23)(7 27 11)(9 16 26)(12 18 15)(19 25 22)
(1 24 11)(2 25 12)(3 26 13)(4 27 14)(5 19 15)(6 20 16)(7 21 17)(8 22 18)(9 23 10)
(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)
G:=sub<Sym(27)| (1,21,14)(2,25,12)(3,16,23)(4,24,17)(5,19,15)(6,10,26)(7,27,11)(8,22,18)(9,13,20), (1,21,14)(2,8,5)(3,10,20)(4,24,17)(6,13,23)(7,27,11)(9,16,26)(12,18,15)(19,25,22), (1,24,11)(2,25,12)(3,26,13)(4,27,14)(5,19,15)(6,20,16)(7,21,17)(8,22,18)(9,23,10), (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)>;
G:=Group( (1,21,14)(2,25,12)(3,16,23)(4,24,17)(5,19,15)(6,10,26)(7,27,11)(8,22,18)(9,13,20), (1,21,14)(2,8,5)(3,10,20)(4,24,17)(6,13,23)(7,27,11)(9,16,26)(12,18,15)(19,25,22), (1,24,11)(2,25,12)(3,26,13)(4,27,14)(5,19,15)(6,20,16)(7,21,17)(8,22,18)(9,23,10), (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) );
G=PermutationGroup([[(1,21,14),(2,25,12),(3,16,23),(4,24,17),(5,19,15),(6,10,26),(7,27,11),(8,22,18),(9,13,20)], [(1,21,14),(2,8,5),(3,10,20),(4,24,17),(6,13,23),(7,27,11),(9,16,26),(12,18,15),(19,25,22)], [(1,24,11),(2,25,12),(3,26,13),(4,27,14),(5,19,15),(6,20,16),(7,21,17),(8,22,18),(9,23,10)], [(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)]])
G:=TransitiveGroup(27,89);
►On 27 points - transitive group 27T98
Generators in S27
(1 7 4)(2 22 18)(3 9 6)(5 25 12)(8 19 15)(10 16 13)(11 17 14)(20 26 23)(21 27 24)
(2 15 25)(3 26 16)(5 18 19)(6 20 10)(8 12 22)(9 23 13)
(1 24 14)(2 25 15)(3 26 16)(4 27 17)(5 19 18)(6 20 10)(7 21 11)(8 22 12)(9 23 13)
(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)
G:=sub<Sym(27)| (1,7,4)(2,22,18)(3,9,6)(5,25,12)(8,19,15)(10,16,13)(11,17,14)(20,26,23)(21,27,24), (2,15,25)(3,26,16)(5,18,19)(6,20,10)(8,12,22)(9,23,13), (1,24,14)(2,25,15)(3,26,16)(4,27,17)(5,19,18)(6,20,10)(7,21,11)(8,22,12)(9,23,13), (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)>;
G:=Group( (1,7,4)(2,22,18)(3,9,6)(5,25,12)(8,19,15)(10,16,13)(11,17,14)(20,26,23)(21,27,24), (2,15,25)(3,26,16)(5,18,19)(6,20,10)(8,12,22)(9,23,13), (1,24,14)(2,25,15)(3,26,16)(4,27,17)(5,19,18)(6,20,10)(7,21,11)(8,22,12)(9,23,13), (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) );
G=PermutationGroup([[(1,7,4),(2,22,18),(3,9,6),(5,25,12),(8,19,15),(10,16,13),(11,17,14),(20,26,23),(21,27,24)], [(2,15,25),(3,26,16),(5,18,19),(6,20,10),(8,12,22),(9,23,13)], [(1,24,14),(2,25,15),(3,26,16),(4,27,17),(5,19,18),(6,20,10),(7,21,11),(8,22,12),(9,23,13)], [(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)]])
G:=TransitiveGroup(27,98);
C33⋊C9 is a maximal subgroup of
C33⋊1C18 C33⋊1D9 C33⋊2D9
51 conjugacy classes
| class | 1 | 3A | ··· | 3H | 3I | ··· | 3AF | 9A | ··· | 9R |
| order | 1 | 3 | ··· | 3 | 3 | ··· | 3 | 9 | ··· | 9 |
| size | 1 | 1 | ··· | 1 | 3 | ··· | 3 | 9 | ··· | 9 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 |
| type | + | ||||||
| image | C1 | C3 | C3 | C9 | He3 | 3- 1+2 | C3≀C3 |
| kernel | C33⋊C9 | C32⋊C9 | C34 | C33 | C32 | C32 | C3 |
| # reps | 1 | 6 | 2 | 18 | 2 | 4 | 18 |
Matrix representation of C33⋊C9 ►in GL4(𝔽19) generated by
| 7 | 0 | 0 | 0 |
| 0 | 11 | 0 | 1 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 8 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 5 | 0 | 0 | 0 |
| 0 | 12 | 1 | 1 |
| 0 | 18 | 0 | 11 |
| 0 | 9 | 0 | 7 |
G:=sub<GL(4,GF(19))| [7,0,0,0,0,11,0,0,0,0,11,0,0,1,11,7],[1,0,0,0,0,1,0,0,0,0,11,0,0,8,11,7],[1,0,0,0,0,11,0,0,0,0,11,0,0,0,0,11],[5,0,0,0,0,12,18,9,0,1,0,0,0,1,11,7] >;
C33⋊C9 in GAP, Magma, Sage, TeX
C_3^3\rtimes C_9
% in TeX
G:=Group("C3^3:C9"); // GroupNames label
G:=SmallGroup(243,13);
// by ID
G=gap.SmallGroup(243,13);
# by ID
G:=PCGroup([5,-3,3,-3,3,-3,135,121,1352]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^3=d^9=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1*c,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations