direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic6, C6⋊Q8, C4.11D6, C6.1C23, C22.8D6, C12.11C22, Dic3.1C22, C3⋊1(C2×Q8), (C2×C4).4S3, (C2×C12).3C2, C2.3(C22×S3), (C2×C6).8C22, (C2×Dic3).3C2, SmallGroup(48,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic6
G = < a,b,c | a2=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C2×Dic6
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ9 | 2 | -2 | -2 | 2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | -2 | -2 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ14 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ15 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | √3 | √3 | -√3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
| ρ16 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -√3 | -√3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ17 | 2 | -2 | 2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -√3 | -√3 | √3 | √3 | symplectic lifted from Dic6, Schur index 2 |
| ρ18 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | √3 | √3 | -√3 | symplectic lifted from Dic6, Schur index 2 |
►Regular action on 48 points
Generators in S48
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(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 34 7 28)(2 33 8 27)(3 32 9 26)(4 31 10 25)(5 30 11 36)(6 29 12 35)(13 46 19 40)(14 45 20 39)(15 44 21 38)(16 43 22 37)(17 42 23 48)(18 41 24 47)
G:=sub<Sym(48)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,34,7,28)(2,33,8,27)(3,32,9,26)(4,31,10,25)(5,30,11,36)(6,29,12,35)(13,46,19,40)(14,45,20,39)(15,44,21,38)(16,43,22,37)(17,42,23,48)(18,41,24,47)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (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), (1,34,7,28)(2,33,8,27)(3,32,9,26)(4,31,10,25)(5,30,11,36)(6,29,12,35)(13,46,19,40)(14,45,20,39)(15,44,21,38)(16,43,22,37)(17,42,23,48)(18,41,24,47) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42)], [(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)], [(1,34,7,28),(2,33,8,27),(3,32,9,26),(4,31,10,25),(5,30,11,36),(6,29,12,35),(13,46,19,40),(14,45,20,39),(15,44,21,38),(16,43,22,37),(17,42,23,48),(18,41,24,47)]])
C2×Dic6 is a maximal subgroup of
C6.SD16 C2.Dic12 C12.47D4 C12⋊2Q8 C42⋊7S3 Dic3.D4 C23.11D6 Dic6⋊C4 C12⋊Q8 D6⋊Q8 C4.D12 C8.D6 C12.48D4 C23.12D6 Dic3⋊Q8 Q8.14D6 C2×S3×Q8 Q8○D12 Q8⋊Dic6
C2×Dic6 is a maximal quotient of
C12⋊2Q8 C12.6Q8 Dic3.D4 C12⋊Q8 C4.Dic6 C12.48D4
Matrix representation of C2×Dic6 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 0 | 12 | 5 |
| 0 | 0 | 0 | 10 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 6 | 2 |
| 0 | 0 | 0 | 1 | 7 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,12,10,0,0,0,5,1],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,2,7] >;
C2×Dic6 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_6 % in TeX
G:=Group("C2xDic6"); // GroupNames label
G:=SmallGroup(48,34);
// by ID
G=gap.SmallGroup(48,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,40,182,42,804]);
// Polycyclic
G:=Group<a,b,c|a^2=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×Dic6 in TeX
Character table of C2×Dic6 in TeX