direct product, metacyclic, supersoluble, monomial, A-group
Aliases: Dic3×C12, C12⋊2C12, C32⋊3C42, C62.16C22, C3⋊(C4×C12), (C3×C12)⋊5C4, C2.2(S3×C12), C6.23(C4×S3), (C2×C12).8C6, C6.7(C2×C12), (C2×C6).41D6, (C2×C12).22S3, (C6×C12).11C2, C22.3(S3×C6), C2.2(C6×Dic3), (C2×Dic3).4C6, (C6×Dic3).8C2, C6.19(C2×Dic3), (C2×C4).6(C3×S3), (C2×C6).6(C2×C6), (C3×C6).19(C2×C4), SmallGroup(144,76)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C12 |
Generators and relations for Dic3×C12
G = < a,b,c | a12=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 104 in 68 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C3×Dic3, C3×C12, C62, C4×Dic3, C4×C12, C6×Dic3, C6×C12, Dic3×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, C3×S3, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C4×Dic3, C4×C12, S3×C12, C6×Dic3, Dic3×C12
►On 48 points
(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 19 5 23 9 15)(2 20 6 24 10 16)(3 21 7 13 11 17)(4 22 8 14 12 18)(25 45 33 41 29 37)(26 46 34 42 30 38)(27 47 35 43 31 39)(28 48 36 44 32 40)
(1 38 23 34)(2 39 24 35)(3 40 13 36)(4 41 14 25)(5 42 15 26)(6 43 16 27)(7 44 17 28)(8 45 18 29)(9 46 19 30)(10 47 20 31)(11 48 21 32)(12 37 22 33)
G:=sub<Sym(48)| (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), (1,19,5,23,9,15)(2,20,6,24,10,16)(3,21,7,13,11,17)(4,22,8,14,12,18)(25,45,33,41,29,37)(26,46,34,42,30,38)(27,47,35,43,31,39)(28,48,36,44,32,40), (1,38,23,34)(2,39,24,35)(3,40,13,36)(4,41,14,25)(5,42,15,26)(6,43,16,27)(7,44,17,28)(8,45,18,29)(9,46,19,30)(10,47,20,31)(11,48,21,32)(12,37,22,33)>;
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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,19,5,23,9,15)(2,20,6,24,10,16)(3,21,7,13,11,17)(4,22,8,14,12,18)(25,45,33,41,29,37)(26,46,34,42,30,38)(27,47,35,43,31,39)(28,48,36,44,32,40), (1,38,23,34)(2,39,24,35)(3,40,13,36)(4,41,14,25)(5,42,15,26)(6,43,16,27)(7,44,17,28)(8,45,18,29)(9,46,19,30)(10,47,20,31)(11,48,21,32)(12,37,22,33) );
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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,19,5,23,9,15),(2,20,6,24,10,16),(3,21,7,13,11,17),(4,22,8,14,12,18),(25,45,33,41,29,37),(26,46,34,42,30,38),(27,47,35,43,31,39),(28,48,36,44,32,40)], [(1,38,23,34),(2,39,24,35),(3,40,13,36),(4,41,14,25),(5,42,15,26),(6,43,16,27),(7,44,17,28),(8,45,18,29),(9,46,19,30),(10,47,20,31),(11,48,21,32),(12,37,22,33)]])
Dic3×C12 is a maximal subgroup of
C3⋊C8⋊Dic3 D12⋊2Dic3 C12.80D12 C12.81D12 C62.6C23 Dic3⋊5Dic6 Dic3⋊6Dic6 Dic3.D12 C62.25C23 C62.29C23 C12.27D12 C12.28D12 Dic3⋊Dic6 C62.37C23 C62.38C23 C62.39C23 C62.44C23 C62.47C23 Dic3⋊4D12 Dic3⋊5D12 C12⋊D12 C12⋊3Dic6 S3×C4×C12
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6O | 12A | ··· | 12H | 12I | ··· | 12T | 12U | ··· | 12AJ |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | C3×S3 | C4×S3 | C3×Dic3 | S3×C6 | S3×C12 |
| kernel | Dic3×C12 | C6×Dic3 | C6×C12 | C4×Dic3 | C3×Dic3 | C3×C12 | C2×Dic3 | C2×C12 | Dic3 | C12 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of Dic3×C12 ►in GL3(𝔽13) generated by
| 9 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 12 | 0 | 0 |
| 0 | 9 | 2 |
| 0 | 0 | 3 |
| 8 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 3 | 1 |
G:=sub<GL(3,GF(13))| [9,0,0,0,2,0,0,0,2],[12,0,0,0,9,0,0,2,3],[8,0,0,0,12,3,0,0,1] >;
Dic3×C12 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{12} % in TeX
G:=Group("Dic3xC12"); // GroupNames label
G:=SmallGroup(144,76);
// by ID
G=gap.SmallGroup(144,76);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,151,3461]);
// Polycyclic
G:=Group<a,b,c|a^12=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations