non-abelian, supersoluble, monomial
Aliases: C3≀C3⋊2C6, C3≀S3⋊1C3, (C3×He3)⋊7C6, (C3×He3)⋊5S3, C33⋊2(C3×S3), He3.4(C3×C6), C33⋊C32⋊2C2, C32.5(S3×C32), He3⋊C2.3C32, C32.21(C32⋊C6), (C3×He3⋊C2)⋊3C3, C3.19(C3×C32⋊C6), SmallGroup(486,126)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C3≀C3⋊C6 |
Generators and relations for C3≀C3⋊C6
G = < a,b,c,d,e | a3=b3=c3=d3=e6=1, ab=ba, cac-1=ab-1, ad=da, eae-1=a-1b, bc=cb, ede-1=bd=db, be=eb, dcd-1=ab-1c, ece-1=c-1 >
Subgroups: 522 in 88 conjugacy classes, 21 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C3×S3, C3×C6, C3×C9, He3, He3, 3- 1+2, C33, C33, C33, He3⋊C2, C2×He3, S3×C32, C3≀C3, C3≀C3, C3×He3, C3×3- 1+2, C3≀S3, S3×He3, C3×He3⋊C2, C33⋊C32, C3≀C3⋊C6
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, C32⋊C6, S3×C32, C3×C32⋊C6, C3≀C3⋊C6
►On 27 points - transitive group 27T150
Generators in S27
(1 6 7)(2 4 8)(3 5 9)(10 22 17)(12 24 19)(14 26 21)
(1 7 6)(2 8 4)(3 9 5)(10 22 17)(11 23 18)(12 24 19)(13 25 20)(14 26 21)(15 27 16)
(1 22 25)(2 26 23)(3 24 27)(4 14 11)(5 12 15)(6 10 13)(7 17 20)(8 21 18)(9 19 16)
(1 4 5)(2 3 7)(6 8 9)(10 26 12)(11 16 20)(13 23 15)(14 19 17)(18 27 25)(21 24 22)
(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)| (1,6,7)(2,4,8)(3,5,9)(10,22,17)(12,24,19)(14,26,21), (1,7,6)(2,8,4)(3,9,5)(10,22,17)(11,23,18)(12,24,19)(13,25,20)(14,26,21)(15,27,16), (1,22,25)(2,26,23)(3,24,27)(4,14,11)(5,12,15)(6,10,13)(7,17,20)(8,21,18)(9,19,16), (1,4,5)(2,3,7)(6,8,9)(10,26,12)(11,16,20)(13,23,15)(14,19,17)(18,27,25)(21,24,22), (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( (1,6,7)(2,4,8)(3,5,9)(10,22,17)(12,24,19)(14,26,21), (1,7,6)(2,8,4)(3,9,5)(10,22,17)(11,23,18)(12,24,19)(13,25,20)(14,26,21)(15,27,16), (1,22,25)(2,26,23)(3,24,27)(4,14,11)(5,12,15)(6,10,13)(7,17,20)(8,21,18)(9,19,16), (1,4,5)(2,3,7)(6,8,9)(10,26,12)(11,16,20)(13,23,15)(14,19,17)(18,27,25)(21,24,22), (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([[(1,6,7),(2,4,8),(3,5,9),(10,22,17),(12,24,19),(14,26,21)], [(1,7,6),(2,8,4),(3,9,5),(10,22,17),(11,23,18),(12,24,19),(13,25,20),(14,26,21),(15,27,16)], [(1,22,25),(2,26,23),(3,24,27),(4,14,11),(5,12,15),(6,10,13),(7,17,20),(8,21,18),(9,19,16)], [(1,4,5),(2,3,7),(6,8,9),(10,26,12),(11,16,20),(13,23,15),(14,19,17),(18,27,25),(21,24,22)], [(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(27,150);
►On 27 points - transitive group 27T168
Generators in S27
(1 3 2)(4 8 6)(10 11 12)(13 15 14)(17 21 19)(22 24 26)
(1 2 3)(4 8 6)(5 9 7)(10 12 11)(13 14 15)(16 20 18)(17 21 19)(22 24 26)(23 25 27)
(1 6 7)(2 4 5)(3 8 9)(10 22 25)(11 26 23)(12 24 27)(13 21 18)(14 19 16)(15 17 20)
(1 10 13)(2 12 14)(3 11 15)(4 22 17)(5 25 20)(6 26 19)(7 23 16)(8 24 21)(9 27 18)
(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)| (1,3,2)(4,8,6)(10,11,12)(13,15,14)(17,21,19)(22,24,26), (1,2,3)(4,8,6)(5,9,7)(10,12,11)(13,14,15)(16,20,18)(17,21,19)(22,24,26)(23,25,27), (1,6,7)(2,4,5)(3,8,9)(10,22,25)(11,26,23)(12,24,27)(13,21,18)(14,19,16)(15,17,20), (1,10,13)(2,12,14)(3,11,15)(4,22,17)(5,25,20)(6,26,19)(7,23,16)(8,24,21)(9,27,18), (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( (1,3,2)(4,8,6)(10,11,12)(13,15,14)(17,21,19)(22,24,26), (1,2,3)(4,8,6)(5,9,7)(10,12,11)(13,14,15)(16,20,18)(17,21,19)(22,24,26)(23,25,27), (1,6,7)(2,4,5)(3,8,9)(10,22,25)(11,26,23)(12,24,27)(13,21,18)(14,19,16)(15,17,20), (1,10,13)(2,12,14)(3,11,15)(4,22,17)(5,25,20)(6,26,19)(7,23,16)(8,24,21)(9,27,18), (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([[(1,3,2),(4,8,6),(10,11,12),(13,15,14),(17,21,19),(22,24,26)], [(1,2,3),(4,8,6),(5,9,7),(10,12,11),(13,14,15),(16,20,18),(17,21,19),(22,24,26),(23,25,27)], [(1,6,7),(2,4,5),(3,8,9),(10,22,25),(11,26,23),(12,24,27),(13,21,18),(14,19,16),(15,17,20)], [(1,10,13),(2,12,14),(3,11,15),(4,22,17),(5,25,20),(6,26,19),(7,23,16),(8,24,21),(9,27,18)], [(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(27,168);
►On 27 points - transitive group 27T191
Generators in S27
(1 5 6)(2 7 8)(3 9 4)(10 17 18)(11 21 16)(12 19 20)(13 22 27)(14 26 25)(15 24 23)
(1 3 2)(4 8 6)(5 9 7)(10 12 11)(13 14 15)(16 18 20)(17 19 21)(22 26 24)(23 27 25)
(4 6 8)(5 9 7)(10 17 20)(11 21 18)(12 19 16)(13 27 24)(14 25 22)(15 23 26)
(1 10 15)(2 11 14)(3 12 13)(4 20 27)(5 17 24)(6 18 23)(7 21 26)(8 16 25)(9 19 22)
(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)| (1,5,6)(2,7,8)(3,9,4)(10,17,18)(11,21,16)(12,19,20)(13,22,27)(14,26,25)(15,24,23), (1,3,2)(4,8,6)(5,9,7)(10,12,11)(13,14,15)(16,18,20)(17,19,21)(22,26,24)(23,27,25), (4,6,8)(5,9,7)(10,17,20)(11,21,18)(12,19,16)(13,27,24)(14,25,22)(15,23,26), (1,10,15)(2,11,14)(3,12,13)(4,20,27)(5,17,24)(6,18,23)(7,21,26)(8,16,25)(9,19,22), (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( (1,5,6)(2,7,8)(3,9,4)(10,17,18)(11,21,16)(12,19,20)(13,22,27)(14,26,25)(15,24,23), (1,3,2)(4,8,6)(5,9,7)(10,12,11)(13,14,15)(16,18,20)(17,19,21)(22,26,24)(23,27,25), (4,6,8)(5,9,7)(10,17,20)(11,21,18)(12,19,16)(13,27,24)(14,25,22)(15,23,26), (1,10,15)(2,11,14)(3,12,13)(4,20,27)(5,17,24)(6,18,23)(7,21,26)(8,16,25)(9,19,22), (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([[(1,5,6),(2,7,8),(3,9,4),(10,17,18),(11,21,16),(12,19,20),(13,22,27),(14,26,25),(15,24,23)], [(1,3,2),(4,8,6),(5,9,7),(10,12,11),(13,14,15),(16,18,20),(17,19,21),(22,26,24),(23,27,25)], [(4,6,8),(5,9,7),(10,17,20),(11,21,18),(12,19,16),(13,27,24),(14,25,22),(15,23,26)], [(1,10,15),(2,11,14),(3,12,13),(4,20,27),(5,17,24),(6,18,23),(7,21,26),(8,16,25),(9,19,22)], [(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(27,191);
34 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | ··· | 3M | 3N | 3O | 3P | 6A | 6B | 6C | ··· | 6J | 9A | ··· | 9F |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 6 | 6 | 6 | ··· | 6 | 9 | ··· | 9 |
| size | 1 | 9 | 1 | 1 | 3 | 3 | 6 | 6 | 6 | 9 | ··· | 9 | 18 | 18 | 18 | 9 | 9 | 27 | ··· | 27 | 18 | ··· | 18 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 9 |
| type | + | + | + | + | |||||||
| image | C1 | C2 | C3 | C3 | C6 | C6 | S3 | C3×S3 | C32⋊C6 | C3×C32⋊C6 | C3≀C3⋊C6 |
| kernel | C3≀C3⋊C6 | C33⋊C32 | C3≀S3 | C3×He3⋊C2 | C3≀C3 | C3×He3 | C3×He3 | C33 | C32 | C3 | C1 |
| # reps | 1 | 1 | 6 | 2 | 6 | 2 | 1 | 8 | 1 | 2 | 4 |
Matrix representation of C3≀C3⋊C6 ►in GL9(𝔽19)
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 8 | 7 | 0 | 1 | 8 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 1 | 8 | 0 | 12 | 1 | 0 | 0 | 0 | 1 |
| 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 |
| 0 | 0 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 7 | 7 | 8 | 8 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 12 | 11 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 0 | 0 |
| 12 | 11 | 0 | 11 | 18 | 18 | 0 | 1 | 0 |
| 18 | 0 | 0 | 8 | 0 | 0 | 4 | 0 | 0 |
| 0 | 12 | 0 | 0 | 18 | 0 | 0 | 9 | 0 |
| 8 | 8 | 1 | 1 | 1 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 12 | 12 | 0 | 0 | 18 | 12 | 12 |
| 12 | 0 | 0 | 7 | 0 | 0 | 1 | 0 | 0 |
| 0 | 8 | 0 | 0 | 11 | 0 | 0 | 7 | 0 |
| 7 | 18 | 7 | 0 | 8 | 1 | 7 | 11 | 18 |
| 8 | 0 | 0 | 12 | 0 | 0 | 6 | 0 | 0 |
| 18 | 18 | 7 | 7 | 7 | 0 | 0 | 0 | 6 |
| 0 | 8 | 0 | 0 | 12 | 0 | 0 | 6 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 8 | 8 | 8 |
| 0 | 0 | 0 | 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 11 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 12 | 1 |
G:=sub<GL(9,GF(19))| [7,0,0,0,0,8,0,0,1,0,11,0,0,0,7,0,0,8,0,0,1,0,0,0,0,0,0,0,0,0,7,0,1,0,0,12,0,0,0,0,11,8,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,1],[11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,11],[0,7,0,7,0,1,0,0,12,0,0,1,7,0,12,0,0,11,11,0,0,7,0,0,0,0,0,0,0,0,8,7,12,0,0,11,0,0,0,8,0,11,0,0,18,0,0,0,9,0,11,1,0,18,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0],[18,0,8,1,0,0,12,0,7,0,12,8,0,7,12,0,8,18,0,0,1,0,0,12,0,0,7,8,0,1,0,0,12,7,0,0,0,18,1,0,0,0,0,11,8,0,0,0,0,0,0,0,0,1,4,0,0,0,0,18,1,0,7,0,9,0,0,0,12,0,7,11,0,0,9,0,0,12,0,0,18],[8,18,0,1,0,0,0,0,0,0,18,8,0,0,0,0,0,0,0,7,0,0,1,0,0,0,0,12,7,0,0,0,0,0,0,0,0,7,12,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,6,0,0,0,0,8,11,0,1,0,0,6,0,0,8,0,0,12,0,6,0,0,0,8,0,11,1] >;
C3≀C3⋊C6 in GAP, Magma, Sage, TeX
C_3\wr C_3\rtimes C_6
% in TeX
G:=Group("C3wrC3:C6"); // GroupNames label
G:=SmallGroup(486,126);
// by ID
G=gap.SmallGroup(486,126);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1520,867,873,8104,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^3=e^6=1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e^-1=a^-1*b,b*c=c*b,e*d*e^-1=b*d=d*b,b*e=e*b,d*c*d^-1=a*b^-1*c,e*c*e^-1=c^-1>;
// generators/relations