direct product, metabelian, supersoluble, monomial
Aliases: C6×D12, C12⋊8D6, C62.28C22, C4⋊2(S3×C6), C6⋊1(C3×D4), (C3×C6)⋊4D4, C3⋊1(C6×D4), (C2×C12)⋊6S3, C12⋊2(C2×C6), (C6×C12)⋊7C2, (C2×C12)⋊3C6, D6⋊1(C2×C6), C32⋊8(C2×D4), (C2×C6).48D6, (C22×S3)⋊2C6, (S3×C6)⋊7C22, (C3×C12)⋊7C22, C6.3(C22×C6), (C3×C6).21C23, C22.10(S3×C6), C6.42(C22×S3), (S3×C2×C6)⋊4C2, C2.4(S3×C2×C6), (C2×C4)⋊2(C3×S3), (C2×C6).13(C2×C6), SmallGroup(144,160)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×D12
G = < a,b,c | a6=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 264 in 116 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C32, C12, C12, D6, D6, C2×C6, C2×C6, C2×D4, C3×S3, C3×C6, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C3×C12, S3×C6, S3×C6, C62, C2×D12, C6×D4, C3×D12, C6×C12, S3×C2×C6, C6×D12
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, S3×C6, C2×D12, C6×D4, C3×D12, S3×C2×C6, C6×D12
►On 48 points
(1 39 5 43 9 47)(2 40 6 44 10 48)(3 41 7 45 11 37)(4 42 8 46 12 38)(13 31 21 27 17 35)(14 32 22 28 18 36)(15 33 23 29 19 25)(16 34 24 30 20 26)
(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)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(12 24)(25 43)(26 42)(27 41)(28 40)(29 39)(30 38)(31 37)(32 48)(33 47)(34 46)(35 45)(36 44)
G:=sub<Sym(48)| (1,39,5,43,9,47)(2,40,6,44,10,48)(3,41,7,45,11,37)(4,42,8,46,12,38)(13,31,21,27,17,35)(14,32,22,28,18,36)(15,33,23,29,19,25)(16,34,24,30,20,26), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(12,24)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,48)(33,47)(34,46)(35,45)(36,44)>;
G:=Group( (1,39,5,43,9,47)(2,40,6,44,10,48)(3,41,7,45,11,37)(4,42,8,46,12,38)(13,31,21,27,17,35)(14,32,22,28,18,36)(15,33,23,29,19,25)(16,34,24,30,20,26), (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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(12,24)(25,43)(26,42)(27,41)(28,40)(29,39)(30,38)(31,37)(32,48)(33,47)(34,46)(35,45)(36,44) );
G=PermutationGroup([[(1,39,5,43,9,47),(2,40,6,44,10,48),(3,41,7,45,11,37),(4,42,8,46,12,38),(13,31,21,27,17,35),(14,32,22,28,18,36),(15,33,23,29,19,25),(16,34,24,30,20,26)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(12,24),(25,43),(26,42),(27,41),(28,40),(29,39),(30,38),(31,37),(32,48),(33,47),(34,46),(35,45),(36,44)]])
C6×D12 is a maximal subgroup of
C12.D12 D12⋊3Dic3 C6.16D24 D12⋊20D6 D12.28D6 C12.27D12 C62.33C23 C62.54C23 C62.55C23 Dic3⋊D12 D12⋊Dic3 D6⋊2D12 C12⋊D12 C62.84C23 C12⋊2D12 D6⋊4D12 D12⋊24D6 S3×C6×D4
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | C3×S3 | D12 | C3×D4 | S3×C6 | S3×C6 | C3×D12 |
| kernel | C6×D12 | C3×D12 | C6×C12 | S3×C2×C6 | C2×D12 | D12 | C2×C12 | C22×S3 | C2×C12 | C3×C6 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 8 | 2 | 4 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 2 | 8 |
Matrix representation of C6×D12 ►in GL3(𝔽13) generated by
| 4 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 3 |
| 12 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 0 | 11 |
| 0 | 6 | 0 |
G:=sub<GL(3,GF(13))| [4,0,0,0,3,0,0,0,3],[12,0,0,0,6,0,0,0,11],[1,0,0,0,0,6,0,11,0] >;
C6×D12 in GAP, Magma, Sage, TeX
C_6\times D_{12} % in TeX
G:=Group("C6xD12"); // GroupNames label
G:=SmallGroup(144,160);
// by ID
G=gap.SmallGroup(144,160);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,506,122,3461]);
// Polycyclic
G:=Group<a,b,c|a^6=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations