non-abelian, supersoluble, monomial
Aliases: (C3×C12)⋊2D6, He3⋊3(C2×D4), C12⋊S3⋊6S3, C32⋊2(S3×D4), C12.86(S32), He3⋊4D4⋊7C2, He3⋊C2⋊2D4, He3⋊2D4⋊2C2, C4⋊2(C32⋊D6), (C4×He3)⋊2C22, C3.2(D6⋊D6), He3⋊3C4⋊2C22, (C2×He3).9C23, C6.83(C2×S32), (C2×C3⋊S3)⋊2D6, (C2×C32⋊D6)⋊2C2, (C4×He3⋊C2)⋊3C2, (C3×C6).9(C22×S3), C2.11(C2×C32⋊D6), (C2×C32⋊C6)⋊2C22, (C2×He3⋊C2).15C22, SmallGroup(432,302)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C3×C12)⋊D6
G = < a,b,c,d | a3=b12=c6=d2=1, ab=ba, cac-1=dad=a-1b4, cbc-1=b-1, bd=db, dcd=c-1 >
Subgroups: 1555 in 205 conjugacy classes, 37 normal (13 characteristic)
C1, C2, C2 [×6], C3, C3 [×3], C4, C4, C22 [×9], S3 [×14], C6, C6 [×9], C2×C4, D4 [×4], C23 [×2], C32 [×2], C32, Dic3 [×3], C12, C12 [×4], D6 [×23], C2×C6 [×5], C2×D4, C3×S3 [×10], C3⋊S3 [×4], C3×C6 [×2], C3×C6, C4×S3 [×3], D12 [×6], C3⋊D4 [×4], C2×C12, C3×D4 [×2], C22×S3 [×6], He3, C3×Dic3 [×3], C3×C12 [×2], C3×C12, S32 [×8], S3×C6 [×7], C2×C3⋊S3 [×4], C2×D12, S3×D4 [×2], C32⋊C6 [×4], He3⋊C2 [×2], C2×He3, C3⋊D12 [×4], S3×C12 [×3], C3×D12 [×2], C12⋊S3 [×2], C2×S32 [×4], He3⋊3C4, C4×He3, C32⋊D6 [×4], C2×C32⋊C6 [×4], C2×He3⋊C2, S3×D12 [×2], He3⋊2D4 [×2], He3⋊4D4 [×2], C4×He3⋊C2, C2×C32⋊D6 [×2], (C3×C12)⋊D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C22×S3 [×2], S32, S3×D4 [×2], C2×S32, C32⋊D6, D6⋊D6, C2×C32⋊D6, (C3×C12)⋊D6
►On 36 points
(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)
(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 28 29 30 31 32 33 34 35 36)
(1 31 15)(2 30 16 12 32 14)(3 29 17 11 33 13)(4 28 18 10 34 24)(5 27 19 9 35 23)(6 26 20 8 36 22)(7 25 21)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)
G:=sub<Sym(36)| (13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,31,15)(2,30,16,12,32,14)(3,29,17,11,33,13)(4,28,18,10,34,24)(5,27,19,9,35,23)(6,26,20,8,36,22)(7,25,21), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,13)(12,14)>;
G:=Group( (13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36), (1,31,15)(2,30,16,12,32,14)(3,29,17,11,33,13)(4,28,18,10,34,24)(5,27,19,9,35,23)(6,26,20,8,36,22)(7,25,21), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,13)(12,14) );
G=PermutationGroup([(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36)], [(1,31,15),(2,30,16,12,32,14),(3,29,17,11,33,13),(4,28,18,10,34,24),(5,27,19,9,35,23),(6,26,20,8,36,22),(7,25,21)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,13),(12,14)])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| 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 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 9 | 9 | 18 | 18 | 18 | 18 | 2 | 6 | 6 | 12 | 2 | 18 | 2 | 6 | 6 | 12 | 18 | 18 | 36 | 36 | 36 | 36 | 2 | 2 | 12 | 12 | 12 | 12 | 18 | 18 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | S32 | S3×D4 | C2×S32 | D6⋊D6 | C32⋊D6 | C2×C32⋊D6 | (C3×C12)⋊D6 |
| kernel | (C3×C12)⋊D6 | He3⋊2D4 | He3⋊4D4 | C4×He3⋊C2 | C2×C32⋊D6 | C12⋊S3 | He3⋊C2 | C3×C12 | C2×C3⋊S3 | C12 | C32 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of (C3×C12)⋊D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 3 | 3 | 0 | 0 | 0 | 0 |
| 10 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 10 | 6 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 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 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[3,10,0,0,0,0,3,6,0,0,0,0,0,0,3,10,0,0,0,0,3,6,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,12,0,0,0,0,12,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0] >;
(C3×C12)⋊D6 in GAP, Magma, Sage, TeX
(C_3\times C_{12})\rtimes D_6 % in TeX
G:=Group("(C3xC12):D6"); // GroupNames label
G:=SmallGroup(432,302);
// by ID
G=gap.SmallGroup(432,302);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,58,571,4037,537,14118,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations