direct product, non-abelian, soluble, monomial
Aliases: C6×S3≀C2, C32⋊(C6×D4), C33⋊2(C2×D4), (C32×C6)⋊1D4, S32⋊(C2×C6), C3⋊S3⋊(C3×D4), (C3×C6)⋊(C3×D4), (S32×C6)⋊9C2, (C2×S32)⋊5C6, C32⋊C4⋊(C2×C6), (C3×C3⋊S3)⋊1D4, (C3×S32)⋊2C22, (C6×C32⋊C4)⋊7C2, (C2×C32⋊C4)⋊3C6, C3⋊S3.1(C22×C6), (C3×C3⋊S3).3C23, (C3×C32⋊C4)⋊4C22, (C6×C3⋊S3).33C22, (C2×C3⋊S3).9(C2×C6), SmallGroup(432,754)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×S3≀C2
G = < a,b,c,d,e | a6=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >
Subgroups: 956 in 192 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, D4, C23, C32, C32, C12, D6, C2×C6, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×C12, C3×D4, C22×S3, C22×C6, C33, C32⋊C4, S32, S32, S3×C6, C2×C3⋊S3, C62, C6×D4, S3×C32, C3×C3⋊S3, C32×C6, S3≀C2, C2×C32⋊C4, C2×S32, S3×C2×C6, C3×C32⋊C4, C3×S32, C3×S32, S3×C3×C6, C6×C3⋊S3, C2×S3≀C2, C3×S3≀C2, C6×C32⋊C4, S32×C6, C6×S3≀C2
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C6×D4, S3≀C2, C2×S3≀C2, C3×S3≀C2, C6×S3≀C2
►On 24 points - transitive group 24T1316
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(7 9 11)(8 10 12)(19 23 21)(20 24 22)
(1 5 3)(2 6 4)(13 15 17)(14 16 18)
(1 21 16 11)(2 22 17 12)(3 23 18 7)(4 24 13 8)(5 19 14 9)(6 20 15 10)
(1 4)(2 5)(3 6)(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)(13 16)(14 17)(15 18)
G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (7,9,11)(8,10,12)(19,23,21)(20,24,22), (1,5,3)(2,6,4)(13,15,17)(14,16,18), (1,21,16,11)(2,22,17,12)(3,23,18,7)(4,24,13,8)(5,19,14,9)(6,20,15,10), (1,4)(2,5)(3,6)(7,20)(8,21)(9,22)(10,23)(11,24)(12,19)(13,16)(14,17)(15,18)>;
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), (7,9,11)(8,10,12)(19,23,21)(20,24,22), (1,5,3)(2,6,4)(13,15,17)(14,16,18), (1,21,16,11)(2,22,17,12)(3,23,18,7)(4,24,13,8)(5,19,14,9)(6,20,15,10), (1,4)(2,5)(3,6)(7,20)(8,21)(9,22)(10,23)(11,24)(12,19)(13,16)(14,17)(15,18) );
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)], [(7,9,11),(8,10,12),(19,23,21),(20,24,22)], [(1,5,3),(2,6,4),(13,15,17),(14,16,18)], [(1,21,16,11),(2,22,17,12),(3,23,18,7),(4,24,13,8),(5,19,14,9),(6,20,15,10)], [(1,4),(2,5),(3,6),(7,20),(8,21),(9,22),(10,23),(11,24),(12,19),(13,16),(14,17),(15,18)]])
G:=TransitiveGroup(24,1316);
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6P | 6Q | 6R | 6S | 6T | 6U | ··· | 6AF | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 6 | 6 | 6 | 6 | 9 | 9 | 1 | 1 | 4 | ··· | 4 | 18 | 18 | 1 | 1 | 4 | ··· | 4 | 6 | ··· | 6 | 9 | 9 | 9 | 9 | 12 | ··· | 12 | 18 | 18 | 18 | 18 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | S3≀C2 | C2×S3≀C2 | C3×S3≀C2 | C6×S3≀C2 |
| kernel | C6×S3≀C2 | C3×S3≀C2 | C6×C32⋊C4 | S32×C6 | C2×S3≀C2 | S3≀C2 | C2×C32⋊C4 | C2×S32 | C3×C3⋊S3 | C32×C6 | C3⋊S3 | C3×C6 | C6 | C3 | C2 | C1 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 |
Matrix representation of C6×S3≀C2 ►in GL4(𝔽7) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 3 | 5 | 5 | 1 |
| 3 | 1 | 1 | 5 |
| 6 | 4 | 6 | 3 |
| 1 | 6 | 6 | 5 |
| 4 | 2 | 5 | 0 |
| 2 | 3 | 5 | 2 |
| 2 | 2 | 6 | 2 |
| 2 | 4 | 3 | 2 |
| 2 | 4 | 0 | 3 |
| 2 | 5 | 1 | 4 |
| 1 | 4 | 2 | 3 |
| 1 | 6 | 3 | 5 |
| 3 | 5 | 5 | 1 |
| 6 | 2 | 1 | 3 |
| 2 | 5 | 6 | 1 |
| 1 | 6 | 6 | 5 |
G:=sub<GL(4,GF(7))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[3,3,6,1,5,1,4,6,5,1,6,6,1,5,3,5],[4,2,2,2,2,3,2,4,5,5,6,3,0,2,2,2],[2,2,1,1,4,5,4,6,0,1,2,3,3,4,3,5],[3,6,2,1,5,2,5,6,5,1,6,6,1,3,1,5] >;
C6×S3≀C2 in GAP, Magma, Sage, TeX
C_6\times S_3\wr C_2
% in TeX
G:=Group("C6xS3wrC2"); // GroupNames label
G:=SmallGroup(432,754);
// by ID
G=gap.SmallGroup(432,754);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,3,365,4037,3036,201,1189,622]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=e^2=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=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
// generators/relations