direct product, non-abelian, supersoluble, monomial
Aliases: C2×C3≀S3, C33⋊7D6, He3⋊(C2×C6), (C2×He3)⋊C6, C3≀C3⋊3C22, (C32×C6)⋊1S3, He3⋊C2⋊1C6, C32.1(S3×C6), C6.15(C32⋊C6), (C2×C3≀C3)⋊2C2, (C3×C6).4(C3×S3), C3.6(C2×C32⋊C6), (C2×He3⋊C2)⋊1C3, SmallGroup(324,68)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C2×C3≀S3 |
Generators and relations for C2×C3≀S3
G = < a,b,c,d,e | a2=b3=c3=d3=e6=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe-1=b-1c, cd=dc, ce=ec, ede-1=b-1d-1 >
Subgroups: 376 in 84 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2, C3, C3, C22, S3, C6, C6, C9, C32, C32, D6, C2×C6, C18, C3×S3, C3×C6, C3×C6, He3, 3- 1+2, C33, S3×C6, C62, He3⋊C2, C2×He3, C2×3- 1+2, S3×C32, C32×C6, C3≀C3, C2×He3⋊C2, S3×C3×C6, C3≀S3, C2×C3≀C3, C2×C3≀S3
Quotients: C1, C2, C3, C22, S3, C6, D6, C2×C6, C3×S3, S3×C6, C32⋊C6, C2×C32⋊C6, C3≀S3, C2×C3≀S3
►On 18 points - transitive group 18T119
(1 2)(3 4)(5 6)(7 15)(8 16)(9 17)(10 18)(11 13)(12 14)
(1 6 4)(2 5 3)(8 10 12)(14 16 18)
(1 4 6)(2 3 5)(7 9 11)(8 10 12)(13 15 17)(14 16 18)
(1 16 11)(2 8 13)(3 10 15)(4 18 7)(5 12 17)(6 14 9)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,15)(8,16)(9,17)(10,18)(11,13)(12,14), (1,6,4)(2,5,3)(8,10,12)(14,16,18), (1,4,6)(2,3,5)(7,9,11)(8,10,12)(13,15,17)(14,16,18), (1,16,11)(2,8,13)(3,10,15)(4,18,7)(5,12,17)(6,14,9), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)>;
G:=Group( (1,2)(3,4)(5,6)(7,15)(8,16)(9,17)(10,18)(11,13)(12,14), (1,6,4)(2,5,3)(8,10,12)(14,16,18), (1,4,6)(2,3,5)(7,9,11)(8,10,12)(13,15,17)(14,16,18), (1,16,11)(2,8,13)(3,10,15)(4,18,7)(5,12,17)(6,14,9), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18) );
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,15),(8,16),(9,17),(10,18),(11,13),(12,14)], [(1,6,4),(2,5,3),(8,10,12),(14,16,18)], [(1,4,6),(2,3,5),(7,9,11),(8,10,12),(13,15,17),(14,16,18)], [(1,16,11),(2,8,13),(3,10,15),(4,18,7),(5,12,17),(6,14,9)], [(1,2),(3,4),(5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)]])
G:=TransitiveGroup(18,119);
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 6A | 6B | 6C | ··· | 6H | 6I | 6J | ··· | 6Y | 6Z | 9A | 9B | 18A | 18B |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | ··· | 6 | 6 | 9 | 9 | 18 | 18 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 3 | ··· | 3 | 6 | 18 | 1 | 1 | 3 | ··· | 3 | 6 | 9 | ··· | 9 | 18 | 18 | 18 | 18 | 18 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | 6 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D6 | C3×S3 | S3×C6 | C3≀S3 | C2×C3≀S3 | C32⋊C6 | C2×C32⋊C6 |
| kernel | C2×C3≀S3 | C3≀S3 | C2×C3≀C3 | C2×He3⋊C2 | He3⋊C2 | C2×He3 | C32×C6 | C33 | C3×C6 | C32 | C2 | C1 | C6 | C3 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 | 12 | 12 | 1 | 1 |
Matrix representation of C2×C3≀S3 ►in GL3(𝔽7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 0 | 1 | 5 |
| 1 | 0 | 2 |
| 6 | 2 | 0 |
| 4 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 4 |
| 4 | 0 | 0 |
| 2 | 0 | 4 |
| 1 | 3 | 3 |
| 1 | 5 | 0 |
| 0 | 2 | 3 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[0,1,6,1,0,2,5,2,0],[4,0,0,0,4,0,0,0,4],[4,2,1,0,0,3,0,4,3],[1,0,0,5,2,1,0,3,0] >;
C2×C3≀S3 in GAP, Magma, Sage, TeX
C_2\times C_3\wr S_3
% in TeX
G:=Group("C2xC3wrS3"); // GroupNames label
G:=SmallGroup(324,68);
// by ID
G=gap.SmallGroup(324,68);
# by ID
G:=PCGroup([6,-2,-2,-3,-3,-3,-3,579,303,5404,382]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^3=e^6=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e^-1=b^-1*c,c*d=d*c,c*e=e*c,e*d*e^-1=b^-1*d^-1>;
// generators/relations