metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊C4, C6.5D4, C6.1Q8, C2.1Dic6, C22.4D6, C3⋊1(C4⋊C4), C2.4(C4×S3), C6.3(C2×C4), (C2×C4).1S3, (C2×C12).1C2, C2.1(C3⋊D4), (C2×C6).4C22, (C2×Dic3).1C2, SmallGroup(48,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊C4
G = < a,b,c | a6=c4=1, b2=a3, bab-1=a-1, ac=ca, cbc-1=a3b >
Character table of Dic3⋊C4
| 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 | i | -i | -i | -1 | i | 1 | 1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | i | -i | i | 1 | -i | -1 | 1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | -i | i | -i | 1 | i | -1 | 1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -1 | -i | 1 | 1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| 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 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ13 | 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 |
| ρ14 | 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 |
| ρ15 | 2 | -2 | 2 | -2 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | -1 | 1 | 1 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ16 | 2 | -2 | 2 | -2 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ17 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-3 | -√-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ18 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-3 | √-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
►Regular action 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 34 4 31)(2 33 5 36)(3 32 6 35)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 38 22 41)(20 37 23 40)(21 42 24 39)
(1 23 11 17)(2 24 12 18)(3 19 7 13)(4 20 8 14)(5 21 9 15)(6 22 10 16)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39)>;
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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39) );
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)], [(1,34,4,31),(2,33,5,36),(3,32,6,35),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,44,16,47),(14,43,17,46),(15,48,18,45),(19,38,22,41),(20,37,23,40),(21,42,24,39)], [(1,23,11,17),(2,24,12,18),(3,19,7,13),(4,20,8,14),(5,21,9,15),(6,22,10,16),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39)]])
Dic3⋊C4 is a maximal subgroup of
C4×Dic6 C12.6Q8 C42⋊2S3 C42⋊3S3 C23.16D6 Dic3.D4 C23.8D6 Dic3⋊4D4 C23.9D6 Dic3⋊D4 Dic6⋊C4 C12⋊Q8 Dic3.Q8 C4.Dic6 S3×C4⋊C4 D6.D4 D6⋊Q8 C4⋊C4⋊S3 C12.48D4 C4×C3⋊D4 C23.28D6 C23.23D6 C23.14D6 Dic3⋊Q8 D6⋊3Q8 Dic9⋊C4 Dic3⋊Dic3 C62.C22 C6.Dic6 CSU2(𝔽3)⋊C4 Q8.Dic6 C24.3D6 SL2(𝔽3).D4 Dic15⋊5C4 C6.Dic10 C30.4Q8 Dic3⋊F5 Dic21⋊C4 C14.Dic6 C42.4Q8 C33⋊(C4⋊C4) C33⋊C4⋊C4 C6.PSU3(𝔽2) C22.2S5
Dic3⋊C4 is a maximal quotient of
C6.Q16 C12.Q8 Dic3⋊C8 C12.53D4 C6.C42 Dic9⋊C4 Dic3⋊Dic3 C62.C22 C6.Dic6 C24.3D6 Dic15⋊5C4 C6.Dic10 C30.4Q8 Dic3⋊F5 Dic21⋊C4 C14.Dic6 C42.4Q8 C33⋊(C4⋊C4) C33⋊C4⋊C4 C6.PSU3(𝔽2)
Matrix representation of Dic3⋊C4 ►in GL3(𝔽13) generated by
| 1 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 1 | 1 |
| 12 | 0 | 0 |
| 0 | 9 | 11 |
| 0 | 2 | 4 |
| 5 | 0 | 0 |
| 0 | 2 | 4 |
| 0 | 9 | 11 |
G:=sub<GL(3,GF(13))| [1,0,0,0,0,1,0,12,1],[12,0,0,0,9,2,0,11,4],[5,0,0,0,2,9,0,4,11] >;
Dic3⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes C_4 % in TeX
G:=Group("Dic3:C4"); // GroupNames label
G:=SmallGroup(48,12);
// by ID
G=gap.SmallGroup(48,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,40,101,26,804]);
// Polycyclic
G:=Group<a,b,c|a^6=c^4=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^3*b>;
// generators/relations
Export
Subgroup lattice of Dic3⋊C4 in TeX
Character table of Dic3⋊C4 in TeX