direct product, metabelian, supersoluble, monomial
Aliases: C3×D6⋊S3, C33⋊4D4, C6.27S32, (S3×C6)⋊1S3, (S3×C6)⋊1C6, D6⋊1(C3×S3), C6.3(S3×C6), C3⋊Dic3⋊5C6, (C3×C6).40D6, C32⋊4(C3×D4), C32⋊11(C3⋊D4), (C32×C6).3C22, (S3×C3×C6)⋊1C2, C2.3(C3×S32), C3⋊2(C3×C3⋊D4), (C3×C6).8(C2×C6), (C3×C3⋊Dic3)⋊7C2, SmallGroup(216,121)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D6⋊S3
G = < a,b,c,d,e | a3=b6=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=d-1 >
Subgroups: 276 in 94 conjugacy classes, 28 normal (12 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, D4, C32, C32, C32, Dic3, C12, D6, C2×C6, C3×S3, C3×C6, C3×C6, C3×C6, C3⋊D4, C3×D4, C33, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, S3×C32, C32×C6, D6⋊S3, C3×C3⋊D4, C3×C3⋊Dic3, S3×C3×C6, C3×D6⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, C3×S3, C3⋊D4, C3×D4, S32, S3×C6, D6⋊S3, C3×C3⋊D4, C3×S32, C3×D6⋊S3
►On 24 points - transitive group 24T546
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 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)
(1 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 19)(14 24)(15 23)(16 22)(17 21)(18 20)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
G:=sub<Sym(24)| (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;
G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,12)(2,11)(3,10)(4,9)(5,8)(6,7)(13,19)(14,24)(15,23)(16,22)(17,21)(18,20), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,19),(14,24),(15,23),(16,22),(17,21),(18,20)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)]])
G:=TransitiveGroup(24,546);
C3×D6⋊S3 is a maximal subgroup of
C33⋊D8 C33⋊7SD16 D6⋊4S32 (S3×C6)⋊D6 (S3×C6).D6 D6.4S32 D6⋊S3⋊S3 C3×S3×C3⋊D4
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3H | 3I | 3J | 3K | 4 | 6A | 6B | 6C | ··· | 6H | 6I | 6J | 6K | 6L | ··· | 6AA | 12A | 12B |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 3 | 3 | 3 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 |
| size | 1 | 1 | 6 | 6 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 18 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 18 | 18 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | S3 | D4 | D6 | C3×S3 | C3⋊D4 | C3×D4 | S3×C6 | C3×C3⋊D4 | S32 | D6⋊S3 | C3×S32 | C3×D6⋊S3 |
| kernel | C3×D6⋊S3 | C3×C3⋊Dic3 | S3×C3×C6 | D6⋊S3 | C3⋊Dic3 | S3×C6 | S3×C6 | C33 | C3×C6 | D6 | C32 | C32 | C6 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 1 | 2 | 4 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×D6⋊S3 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 1 |
| 5 | 4 | 6 | 0 |
| 1 | 6 | 5 | 5 |
| 3 | 3 | 2 | 0 |
| 3 | 3 | 5 | 0 |
| 6 | 0 | 4 | 6 |
| 6 | 1 | 3 | 2 |
| 0 | 0 | 0 | 1 |
| 2 | 6 | 5 | 6 |
| 4 | 3 | 1 | 3 |
| 1 | 1 | 2 | 5 |
| 1 | 6 | 3 | 5 |
| 1 | 4 | 2 | 0 |
| 5 | 3 | 3 | 1 |
| 4 | 6 | 0 | 5 |
| 3 | 3 | 2 | 3 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,5,1,3,0,4,6,3,3,6,5,2,1,0,5,0],[3,6,6,0,3,0,1,0,5,4,3,0,0,6,2,1],[2,4,1,1,6,3,1,6,5,1,2,3,6,3,5,5],[1,5,4,3,4,3,6,3,2,3,0,2,0,1,5,3] >;
C3×D6⋊S3 in GAP, Magma, Sage, TeX
C_3\times D_6\rtimes S_3
% in TeX
G:=Group("C3xD6:S3"); // GroupNames label
G:=SmallGroup(216,121);
// by ID
G=gap.SmallGroup(216,121);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,730,5189]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^3*c,e*d*e=d^-1>;
// generators/relations