direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×Dic3, C3⋊C42, C12⋊2C4, C22.3D6, C2.2(C4×S3), C6.7(C2×C4), (C2×C4).6S3, (C2×C12).6C2, (C2×C6).3C22, C2.2(C2×Dic3), (C2×Dic3).4C2, SmallGroup(48,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — C4×Dic3 |
Generators and relations for C4×Dic3
G = < a,b,c | a4=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C4×Dic3
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 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 | 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 | 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 | 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 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | i | -1 | 1 | -1 | 1 | -i | i | -i | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | i | -i | -i | i | i | -i | -i | i | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | i | i | -i | -i | i | i | -i | 1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | i | -i | i | -i | -i | 1 | -1 | 1 | -1 | i | -i | i | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -1 | i | -i | -i | i | -1 | 1 | 1 | -1 | 1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | i | i | i | -i | -i | -i | -i | i | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -i | -i | i | i | i | i | -i | 1 | -1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | i | 1 | -1 | 1 | -1 | -i | i | -i | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ13 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | -1 | -i | i | i | -i | -1 | 1 | 1 | -1 | 1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ14 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | i | -i | -1 | 1 | -1 | 1 | i | -i | i | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ15 | 1 | -1 | -1 | 1 | 1 | -i | -i | i | i | 1 | -i | i | i | -i | 1 | -1 | -1 | -1 | 1 | -1 | -i | i | -i | i | linear of order 4 |
| ρ16 | 1 | -1 | -1 | 1 | 1 | i | i | -i | -i | 1 | i | -i | -i | i | 1 | -1 | -1 | -1 | 1 | -1 | i | -i | i | -i | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ18 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ19 | 2 | 2 | -2 | -2 | -1 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ20 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ21 | 2 | -2 | 2 | -2 | -1 | -2i | 2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ22 | 2 | -2 | -2 | 2 | -1 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -i | i | -i | i | complex lifted from C4×S3 |
| ρ23 | 2 | -2 | 2 | -2 | -1 | 2i | -2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ24 | 2 | -2 | -2 | 2 | -1 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | i | -i | i | -i | complex lifted from C4×S3 |
►Regular action on 48 points
Generators in S48
(1 27 16 24)(2 28 17 19)(3 29 18 20)(4 30 13 21)(5 25 14 22)(6 26 15 23)(7 38 45 33)(8 39 46 34)(9 40 47 35)(10 41 48 36)(11 42 43 31)(12 37 44 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)(37 38 39 40 41 42)(43 44 45 46 47 48)
(1 38 4 41)(2 37 5 40)(3 42 6 39)(7 21 10 24)(8 20 11 23)(9 19 12 22)(13 36 16 33)(14 35 17 32)(15 34 18 31)(25 47 28 44)(26 46 29 43)(27 45 30 48)
G:=sub<Sym(48)| (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,38,45,33)(8,39,46,34)(9,40,47,35)(10,41,48,36)(11,42,43,31)(12,37,44,32), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,21,10,24)(8,20,11,23)(9,19,12,22)(13,36,16,33)(14,35,17,32)(15,34,18,31)(25,47,28,44)(26,46,29,43)(27,45,30,48)>;
G:=Group( (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,38,45,33)(8,39,46,34)(9,40,47,35)(10,41,48,36)(11,42,43,31)(12,37,44,32), (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,21,10,24)(8,20,11,23)(9,19,12,22)(13,36,16,33)(14,35,17,32)(15,34,18,31)(25,47,28,44)(26,46,29,43)(27,45,30,48) );
G=PermutationGroup([[(1,27,16,24),(2,28,17,19),(3,29,18,20),(4,30,13,21),(5,25,14,22),(6,26,15,23),(7,38,45,33),(8,39,46,34),(9,40,47,35),(10,41,48,36),(11,42,43,31),(12,37,44,32)], [(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),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,38,4,41),(2,37,5,40),(3,42,6,39),(7,21,10,24),(8,20,11,23),(9,19,12,22),(13,36,16,33),(14,35,17,32),(15,34,18,31),(25,47,28,44),(26,46,29,43),(27,45,30,48)]])
C4×Dic3 is a maximal subgroup of
Dic3⋊C8 C24⋊C4 D12⋊C4 Q8⋊3Dic3 S3×C42 C42⋊2S3 C23.16D6 C23.8D6 Dic3⋊4D4 C23.11D6 Dic6⋊C4 C12⋊Q8 Dic3.Q8 C4.Dic6 C4⋊C4⋊7S3 Dic3⋊5D4 C4⋊C4⋊S3 C23.26D6 C23.12D6 C12⋊3D4 Dic3⋊Q8 C12.23D4 C4.A4⋊C4
C4×Dic3 is a maximal quotient of
C42.S3 C24⋊C4 C6.C42
Matrix representation of C4×Dic3 ►in GL3(𝔽13) generated by
| 8 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 1 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 1 | 1 |
| 12 | 0 | 0 |
| 0 | 0 | 8 |
| 0 | 8 | 0 |
G:=sub<GL(3,GF(13))| [8,0,0,0,12,0,0,0,12],[1,0,0,0,0,1,0,12,1],[12,0,0,0,0,8,0,8,0] >;
C4×Dic3 in GAP, Magma, Sage, TeX
C_4\times {\rm Dic}_3 % in TeX
G:=Group("C4xDic3"); // GroupNames label
G:=SmallGroup(48,11);
// by ID
G=gap.SmallGroup(48,11);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,20,46,804]);
// Polycyclic
G:=Group<a,b,c|a^4=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C4×Dic3 in TeX
Character table of C4×Dic3 in TeX