direct product, metabelian, supersoluble, monomial
Aliases: S3×C3≀C3, C34⋊3C6, (S3×He3)⋊1C3, He3⋊6(C3×S3), (C3×He3)⋊5C6, C3.5(S3×He3), (S3×C33)⋊1C3, C33⋊16(C3×S3), (C3×S3).2He3, C33.25(C3×C6), C32.14(C2×He3), (S3×C32).1C32, C32.22(S3×C32), (C3×3- 1+2)⋊6C6, 3- 1+2⋊5(C3×S3), (S3×3- 1+2)⋊1C3, C3⋊(C2×C3≀C3), (C3×C3≀C3)⋊5C2, SmallGroup(486,117)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C3≀C3
G = < a,b,c,d,e,f | a3=b2=c3=d3=e3=f3=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ece-1=cd-1, cf=fc, de=ed, df=fd, fef-1=cd-1e >
Subgroups: 542 in 123 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3, C3, S3, C6, C9, C32, C32, C18, C3×S3, C3×S3, C3×C6, C3×C9, He3, He3, 3- 1+2, 3- 1+2, C33, C33, S3×C9, C2×He3, C2×3- 1+2, S3×C32, S3×C32, C32×C6, C3≀C3, C3≀C3, C3×He3, C3×3- 1+2, C34, C2×C3≀C3, S3×He3, S3×3- 1+2, S3×C33, C3×C3≀C3, S3×C3≀C3
Quotients: C1, C2, C3, S3, C6, C32, C3×S3, C3×C6, He3, C2×He3, S3×C32, C3≀C3, C2×C3≀C3, S3×He3, S3×C3≀C3
►On 18 points - transitive group 18T163
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)
(1 11)(2 10)(3 12)(4 14)(5 13)(6 15)(7 17)(8 16)(9 18)
(1 3 2)(7 8 9)(10 11 12)(16 18 17)
(1 3 2)(4 6 5)(7 9 8)(10 11 12)(13 14 15)(16 17 18)
(1 4 9)(2 5 7)(3 6 8)(10 13 17)(11 14 18)(12 15 16)
(7 9 8)(16 17 18)
G:=sub<Sym(18)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11)(2,10)(3,12)(4,14)(5,13)(6,15)(7,17)(8,16)(9,18), (1,3,2)(7,8,9)(10,11,12)(16,18,17), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18), (1,4,9)(2,5,7)(3,6,8)(10,13,17)(11,14,18)(12,15,16), (7,9,8)(16,17,18)>;
G:=Group( (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18), (1,11)(2,10)(3,12)(4,14)(5,13)(6,15)(7,17)(8,16)(9,18), (1,3,2)(7,8,9)(10,11,12)(16,18,17), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18), (1,4,9)(2,5,7)(3,6,8)(10,13,17)(11,14,18)(12,15,16), (7,9,8)(16,17,18) );
G=PermutationGroup([[(1,2,3),(4,5,6),(7,8,9),(10,11,12),(13,14,15),(16,17,18)], [(1,11),(2,10),(3,12),(4,14),(5,13),(6,15),(7,17),(8,16),(9,18)], [(1,3,2),(7,8,9),(10,11,12),(16,18,17)], [(1,3,2),(4,6,5),(7,9,8),(10,11,12),(13,14,15),(16,17,18)], [(1,4,9),(2,5,7),(3,6,8),(10,13,17),(11,14,18),(12,15,16)], [(7,9,8),(16,17,18)]])
G:=TransitiveGroup(18,163);
►On 27 points - transitive group 27T207
(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)
(2 3)(5 6)(8 9)(11 12)(14 15)(17 18)(20 21)(23 24)(26 27)
(1 4 7)(2 5 8)(3 6 9)(19 25 22)(20 26 23)(21 27 24)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)
(1 10 22)(2 11 23)(3 12 24)(4 13 25)(5 14 26)(6 15 27)(7 16 19)(8 17 20)(9 18 21)
(19 22 25)(20 23 26)(21 24 27)
G:=sub<Sym(27)| (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), (2,3)(5,6)(8,9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27), (1,4,7)(2,5,8)(3,6,9)(19,25,22)(20,26,23)(21,27,24), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (1,10,22)(2,11,23)(3,12,24)(4,13,25)(5,14,26)(6,15,27)(7,16,19)(8,17,20)(9,18,21), (19,22,25)(20,23,26)(21,24,27)>;
G:=Group( (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), (2,3)(5,6)(8,9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27), (1,4,7)(2,5,8)(3,6,9)(19,25,22)(20,26,23)(21,27,24), (1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27), (1,10,22)(2,11,23)(3,12,24)(4,13,25)(5,14,26)(6,15,27)(7,16,19)(8,17,20)(9,18,21), (19,22,25)(20,23,26)(21,24,27) );
G=PermutationGroup([[(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)], [(2,3),(5,6),(8,9),(11,12),(14,15),(17,18),(20,21),(23,24),(26,27)], [(1,4,7),(2,5,8),(3,6,9),(19,25,22),(20,26,23),(21,27,24)], [(1,4,7),(2,5,8),(3,6,9),(10,13,16),(11,14,17),(12,15,18),(19,22,25),(20,23,26),(21,24,27)], [(1,10,22),(2,11,23),(3,12,24),(4,13,25),(5,14,26),(6,15,27),(7,16,19),(8,17,20),(9,18,21)], [(19,22,25),(20,23,26),(21,24,27)]])
G:=TransitiveGroup(27,207);
51 conjugacy classes
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | ··· | 3M | 3N | ··· | 3U | 3V | 3W | 3X | 3Y | 6A | 6B | 6C | ··· | 6J | 6K | 6L | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 18A | 18B | 18C | 18D |
| order | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 |
| size | 1 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | 9 | 18 | 18 | 3 | 3 | 9 | ··· | 9 | 27 | 27 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 18 | 27 | 27 | 27 | 27 |
51 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C3 | C3 | C3 | C6 | C6 | C6 | S3 | C3×S3 | C3×S3 | C3×S3 | He3 | C2×He3 | C3≀C3 | C2×C3≀C3 | S3×He3 | S3×C3≀C3 |
| kernel | S3×C3≀C3 | C3×C3≀C3 | S3×He3 | S3×3- 1+2 | S3×C33 | C3×He3 | C3×3- 1+2 | C34 | C3≀C3 | He3 | 3- 1+2 | C33 | C3×S3 | C32 | S3 | C3 | C3 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 6 | 6 | 2 | 6 |
Matrix representation of S3×C3≀C3 ►in GL5(𝔽19)
| 18 | 1 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 18 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 18 |
| 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 | 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 | 11 | 9 | 0 |
| 0 | 0 | 12 | 8 | 7 |
| 0 | 0 | 11 | 8 | 0 |
| 7 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 7 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(19))| [18,18,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,18,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,18],[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],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,11,12,11,0,0,9,8,8,0,0,0,7,0],[7,0,0,0,0,0,7,0,0,0,0,0,1,1,0,0,0,0,7,0,0,0,0,0,1] >;
S3×C3≀C3 in GAP, Magma, Sage, TeX
S_3\times C_3\wr C_3
% in TeX
G:=Group("S3xC3wrC3"); // GroupNames label
G:=SmallGroup(486,117);
// by ID
G=gap.SmallGroup(486,117);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,224,303,11669]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^3=d^3=e^3=f^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,e*c*e^-1=c*d^-1,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d^-1*e>;
// generators/relations