direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C6.D6, D6.8S32, Dic3⋊5S32, C3⋊Dic3⋊12D6, (S3×Dic3)⋊6S3, (C3×Dic3)⋊8D6, (S3×C6).18D6, C33⋊3(C22×C4), (C32×C6).2C23, (C32×Dic3)⋊10C22, C2.2S33, C3⋊1(C4×S32), C6.2(C2×S32), (S3×C3⋊S3)⋊1C4, C3⋊S3⋊4(C4×S3), C32⋊6(S3×C2×C4), (C3×S3)⋊1(C4×S3), (C2×C3⋊S3).28D6, (C3×S3×Dic3)⋊10C2, C3⋊1(C2×C6.D6), C33⋊9(C2×C4)⋊10C2, C33⋊8(C2×C4)⋊10C2, (S3×C3×C6).2C22, (S3×C32)⋊3(C2×C4), C33⋊C2⋊2(C2×C4), (C3×C6.D6)⋊9C2, (C6×C3⋊S3).15C22, (C3×C6).51(C22×S3), (C3×C3⋊Dic3)⋊9C22, (C2×C33⋊C2).1C22, (C2×S3×C3⋊S3).2C2, (C3×C3⋊S3)⋊2(C2×C4), SmallGroup(432,595)
Series: Derived ►Chief ►Lower central ►Upper central
| C33 — S3×C6.D6 |
Generators and relations for S3×C6.D6
G = < a,b,c,d,e | a3=b2=c6=e2=1, d6=c3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d5 >
Subgroups: 1788 in 290 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, S3, C6, C6, C6, C2×C4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C22×C4, C3×S3, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C22×S3, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, S3×C32, C3×C3⋊S3, C33⋊C2, C32×C6, S3×Dic3, S3×Dic3, C6.D6, C6.D6, S3×C12, C6×Dic3, C4×C3⋊S3, C2×S32, C22×C3⋊S3, C32×Dic3, C3×C3⋊Dic3, S3×C3⋊S3, S3×C3×C6, C6×C3⋊S3, C2×C33⋊C2, C4×S32, C2×C6.D6, C3×S3×Dic3, C3×C6.D6, C33⋊8(C2×C4), C33⋊9(C2×C4), C2×S3×C3⋊S3, S3×C6.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, S32, S3×C2×C4, C6.D6, C2×S32, C4×S32, C2×C6.D6, S33, S3×C6.D6
►On 24 points - transitive group 24T1298
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 23 21 19 17 15)(14 16 18 20 22 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)
(1 15)(2 20)(3 13)(4 18)(5 23)(6 16)(7 21)(8 14)(9 19)(10 24)(11 17)(12 22)
G:=sub<Sym(24)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,23)(6,16)(7,21)(8,14)(9,19)(10,24)(11,17)(12,22)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18), (1,11,9,7,5,3)(2,4,6,8,10,12)(13,23,21,19,17,15)(14,16,18,20,22,24), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,15)(2,20)(3,13)(4,18)(5,23)(6,16)(7,21)(8,14)(9,19)(10,24)(11,17)(12,22) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18)], [(1,11,9,7,5,3),(2,4,6,8,10,12),(13,23,21,19,17,15),(14,16,18,20,22,24)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,15),(2,20),(3,13),(4,18),(5,23),(6,16),(7,21),(8,14),(9,19),(10,24),(11,17),(12,22)]])
G:=TransitiveGroup(24,1298);
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | 12N | 12O | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 3 | 3 | 9 | 9 | 27 | 27 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 3 | 3 | 3 | 3 | 9 | 9 | 9 | 9 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 12 | 12 | 18 | 18 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 18 | 18 | 18 | 18 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | D6 | D6 | D6 | D6 | C4×S3 | C4×S3 | S32 | S32 | C6.D6 | C2×S32 | C4×S32 | S33 | S3×C6.D6 |
| kernel | S3×C6.D6 | C3×S3×Dic3 | C3×C6.D6 | C33⋊8(C2×C4) | C33⋊9(C2×C4) | C2×S3×C3⋊S3 | S3×C3⋊S3 | S3×Dic3 | C6.D6 | C3×Dic3 | C3⋊Dic3 | S3×C6 | C2×C3⋊S3 | C3×S3 | C3⋊S3 | Dic3 | D6 | S3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 1 | 4 | 2 | 2 | 1 | 8 | 4 | 2 | 1 | 2 | 3 | 4 | 1 | 1 |
Matrix representation of S3×C6.D6 ►in GL6(𝔽13)
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;
S3×C6.D6 in GAP, Magma, Sage, TeX
S_3\times C_6.D_6
% in TeX
G:=Group("S3xC6.D6"); // GroupNames label
G:=SmallGroup(432,595);
// by ID
G=gap.SmallGroup(432,595);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,58,298,2028,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^6=e^2=1,d^6=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^5>;
// generators/relations