non-abelian, supersoluble, monomial
Aliases: C34⋊5S3, C3⋊C3≀S3, C3≀C3⋊3S3, He3⋊5(C3×S3), C33⋊5(C3⋊S3), (C3×He3)⋊10C6, He3⋊5S3⋊3C3, C33.43(C3×S3), C3.13(He3⋊4S3), C32.11(C32⋊C6), (C3×C3≀C3)⋊6C2, C32.2(C3×C3⋊S3), SmallGroup(486,166)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×He3 — C34⋊5S3 |
Generators and relations for C34⋊5S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, ac=ca, ad=da, eae-1=ac-1, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=cd=dc, fcf=c-1d, de=ed, df=fd, fef=e-1 >
Subgroups: 956 in 144 conjugacy classes, 20 normal (12 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C32, C3×S3, C3⋊S3, C3×C6, C3×C9, He3, He3, 3- 1+2, C33, C33, He3⋊C2, S3×C32, C3×C3⋊S3, C3≀C3, C3≀C3, C3×He3, C3×3- 1+2, C34, C3≀S3, He3⋊5S3, C32×C3⋊S3, C3×C3≀C3, C34⋊5S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, C3≀S3, He3⋊4S3, C34⋊5S3
►On 18 points - transitive group 18T167
Generators in S18
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 2 3)(4 5 6)(7 9 8)(10 12 11)(13 15 14)(16 17 18)
(4 6 5)(7 9 8)(10 12 11)(16 18 17)
(1 3 2)(4 6 5)(7 8 9)(10 12 11)(13 15 14)(16 17 18)
(1 4 8)(2 5 7)(3 6 9)(10 17 13)(11 16 14)(12 18 15)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,2,3)(4,5,6)(7,9,8)(10,12,11)(13,15,14)(16,17,18), (4,6,5)(7,9,8)(10,12,11)(16,18,17), (1,3,2)(4,6,5)(7,8,9)(10,12,11)(13,15,14)(16,17,18), (1,4,8)(2,5,7)(3,6,9)(10,17,13)(11,16,14)(12,18,15), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,2,3)(4,5,6)(7,9,8)(10,12,11)(13,15,14)(16,17,18), (4,6,5)(7,9,8)(10,12,11)(16,18,17), (1,3,2)(4,6,5)(7,8,9)(10,12,11)(13,15,14)(16,17,18), (1,4,8)(2,5,7)(3,6,9)(10,17,13)(11,16,14)(12,18,15), (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,2,3),(4,5,6),(7,9,8),(10,12,11),(13,15,14),(16,17,18)], [(4,6,5),(7,9,8),(10,12,11),(16,18,17)], [(1,3,2),(4,6,5),(7,8,9),(10,12,11),(13,15,14),(16,17,18)], [(1,4,8),(2,5,7),(3,6,9),(10,17,13),(11,16,14),(12,18,15)], [(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18)]])
G:=TransitiveGroup(18,167);
►On 27 points - transitive group 27T194
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 9 5)(2 7 6)(3 8 4)(10 13 18)(11 14 16)(12 15 17)(19 23 27)(20 24 25)(21 22 26)
(1 7 4)(2 8 5)(3 9 6)(10 13 18)(11 14 16)(12 15 17)(19 20 21)(22 23 24)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 14 17)(11 15 18)(12 13 16)(19 25 22)(20 26 23)(21 27 24)
(1 10 26)(2 17 20)(3 14 23)(4 11 19)(5 18 22)(6 15 25)(7 12 24)(8 16 27)(9 13 21)
(4 8)(5 9)(6 7)(10 26)(11 27)(12 25)(13 22)(14 23)(15 24)(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), (1,9,5)(2,7,6)(3,8,4)(10,13,18)(11,14,16)(12,15,17)(19,23,27)(20,24,25)(21,22,26), (1,7,4)(2,8,5)(3,9,6)(10,13,18)(11,14,16)(12,15,17)(19,20,21)(22,23,24)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,14,17)(11,15,18)(12,13,16)(19,25,22)(20,26,23)(21,27,24), (1,10,26)(2,17,20)(3,14,23)(4,11,19)(5,18,22)(6,15,25)(7,12,24)(8,16,27)(9,13,21), (4,8)(5,9)(6,7)(10,26)(11,27)(12,25)(13,22)(14,23)(15,24)(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), (1,9,5)(2,7,6)(3,8,4)(10,13,18)(11,14,16)(12,15,17)(19,23,27)(20,24,25)(21,22,26), (1,7,4)(2,8,5)(3,9,6)(10,13,18)(11,14,16)(12,15,17)(19,20,21)(22,23,24)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,14,17)(11,15,18)(12,13,16)(19,25,22)(20,26,23)(21,27,24), (1,10,26)(2,17,20)(3,14,23)(4,11,19)(5,18,22)(6,15,25)(7,12,24)(8,16,27)(9,13,21), (4,8)(5,9)(6,7)(10,26)(11,27)(12,25)(13,22)(14,23)(15,24)(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)], [(1,9,5),(2,7,6),(3,8,4),(10,13,18),(11,14,16),(12,15,17),(19,23,27),(20,24,25),(21,22,26)], [(1,7,4),(2,8,5),(3,9,6),(10,13,18),(11,14,16),(12,15,17),(19,20,21),(22,23,24),(25,26,27)], [(1,3,2),(4,6,5),(7,9,8),(10,14,17),(11,15,18),(12,13,16),(19,25,22),(20,26,23),(21,27,24)], [(1,10,26),(2,17,20),(3,14,23),(4,11,19),(5,18,22),(6,15,25),(7,12,24),(8,16,27),(9,13,21)], [(4,8),(5,9),(6,7),(10,26),(11,27),(12,25),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)]])
G:=TransitiveGroup(27,194);
►On 27 points - transitive group 27T195
Generators in S27
(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 7 4)(2 8 5)(3 9 6)(10 18 15)(11 16 13)(12 17 14)(19 25 22)(20 26 23)(21 27 24)
(1 2 3)(4 5 6)(7 8 9)(19 20 21)(22 23 24)(25 26 27)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)(25 26 27)
(1 12 20)(2 11 19)(3 10 21)(4 14 23)(5 13 22)(6 15 24)(7 17 26)(8 16 25)(9 18 27)
(4 7)(5 8)(6 9)(10 21)(11 19)(12 20)(13 25)(14 26)(15 27)(16 22)(17 23)(18 24)
G:=sub<Sym(27)| (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,7,4)(2,8,5)(3,9,6)(10,18,15)(11,16,13)(12,17,14)(19,25,22)(20,26,23)(21,27,24), (1,2,3)(4,5,6)(7,8,9)(19,20,21)(22,23,24)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,12,20)(2,11,19)(3,10,21)(4,14,23)(5,13,22)(6,15,24)(7,17,26)(8,16,25)(9,18,27), (4,7)(5,8)(6,9)(10,21)(11,19)(12,20)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24)>;
G:=Group( (10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,7,4)(2,8,5)(3,9,6)(10,18,15)(11,16,13)(12,17,14)(19,25,22)(20,26,23)(21,27,24), (1,2,3)(4,5,6)(7,8,9)(19,20,21)(22,23,24)(25,26,27), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27), (1,12,20)(2,11,19)(3,10,21)(4,14,23)(5,13,22)(6,15,24)(7,17,26)(8,16,25)(9,18,27), (4,7)(5,8)(6,9)(10,21)(11,19)(12,20)(13,25)(14,26)(15,27)(16,22)(17,23)(18,24) );
G=PermutationGroup([[(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(1,7,4),(2,8,5),(3,9,6),(10,18,15),(11,16,13),(12,17,14),(19,25,22),(20,26,23),(21,27,24)], [(1,2,3),(4,5,6),(7,8,9),(19,20,21),(22,23,24),(25,26,27)], [(1,3,2),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,17,18),(19,20,21),(22,23,24),(25,26,27)], [(1,12,20),(2,11,19),(3,10,21),(4,14,23),(5,13,22),(6,15,24),(7,17,26),(8,16,25),(9,18,27)], [(4,7),(5,8),(6,9),(10,21),(11,19),(12,20),(13,25),(14,26),(15,27),(16,22),(17,23),(18,24)]])
G:=TransitiveGroup(27,195);
39 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | ··· | 3K | 3L | ··· | 3T | 3U | 3V | 3W | 6A | ··· | 6H | 9A | ··· | 9F |
| order | 1 | 2 | 3 | 3 | 3 | 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 | 18 | 18 | 18 | 27 | ··· | 27 | 18 | ··· | 18 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 6 | 6 |
| type | + | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | S3 | S3 | C3×S3 | C3×S3 | C3≀S3 | C32⋊C6 | C34⋊5S3 |
| kernel | C34⋊5S3 | C3×C3≀C3 | He3⋊5S3 | C3×He3 | C3≀C3 | C34 | He3 | C33 | C3 | C32 | C1 |
| # reps | 1 | 1 | 2 | 2 | 3 | 1 | 6 | 2 | 12 | 3 | 6 |
Matrix representation of C34⋊5S3 ►in GL5(𝔽19)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 0 | 18 | 0 | 0 | 0 |
| 1 | 18 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11,0,0,0,0,0,11],[0,1,0,0,0,18,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0],[0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
C34⋊5S3 in GAP, Magma, Sage, TeX
C_3^4\rtimes_5S_3
% in TeX
G:=Group("C3^4:5S3"); // GroupNames label
G:=SmallGroup(486,166);
// by ID
G=gap.SmallGroup(486,166);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,218,548,867,8104,1096]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*c^-1,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=b^-1,e*c*e^-1=c*d=d*c,f*c*f=c^-1*d,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations