metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊Dic3, C12⋊1C4, C6.4D4, C6.2Q8, C2.1D12, C2.2Dic6, C22.5D6, C3⋊2(C4⋊C4), (C2×C4).3S3, C6.8(C2×C4), (C2×C12).2C2, (C2×C6).5C22, C2.4(C2×Dic3), (C2×Dic3).2C2, SmallGroup(48,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊Dic3
G = < a,b,c | a4=b6=1, c2=b3, ab=ba, cac-1=a-1, cbc-1=b-1 >
Character table of C4⋊Dic3
| 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 | -i | -i | i | i | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | i | i | -i | -i | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -i | i | i | -i | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | i | -i | -i | i | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| 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 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ12 | 2 | 2 | -2 | -2 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ13 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | 2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 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 |
| 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 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | symplectic lifted from Dic3, 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
(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 33 45 38)(8 34 46 39)(9 35 47 40)(10 36 48 41)(11 31 43 42)(12 32 44 37)
(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 33 4 36)(2 32 5 35)(3 31 6 34)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)
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)(7,33,45,38)(8,34,46,39)(9,35,47,40)(10,36,48,41)(11,31,43,42)(12,32,44,37), (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), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45)>;
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)(7,33,45,38)(8,34,46,39)(9,35,47,40)(10,36,48,41)(11,31,43,42)(12,32,44,37), (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), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45) );
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),(7,33,45,38),(8,34,46,39),(9,35,47,40),(10,36,48,41),(11,31,43,42),(12,32,44,37)], [(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)], [(1,33,4,36),(2,32,5,35),(3,31,6,34),(7,30,10,27),(8,29,11,26),(9,28,12,25),(13,41,16,38),(14,40,17,37),(15,39,18,42),(19,44,22,47),(20,43,23,46),(21,48,24,45)]])
C4⋊Dic3 is a maximal subgroup of
C6.Q16 C12.Q8 C2.Dic12 C8⋊Dic3 C24⋊1C4 C2.D24 D4⋊Dic3 Q8⋊2Dic3 C4×Dic6 C12⋊2Q8 C12.6Q8 C4×D12 Dic3.D4 C23.8D6 C23.9D6 C23.21D6 C12⋊Q8 Dic3.Q8 C4.Dic6 S3×C4⋊C4 C4⋊C4⋊7S3 C4.D12 C4⋊C4⋊S3 C12.48D4 C23.26D6 C12⋊7D4 D4×Dic3 D6⋊3D4 Q8×Dic3 D6⋊3Q8 C4⋊Dic9 Dic3⋊Dic3 C12⋊Dic3 Q8.D12 SL2(𝔽3)⋊Q8 C24.4D6 (C2×C4).S4 C30.Q8 C60⋊5C4 C60⋊C4 C42.Q8 C84⋊C4 C6.S3≀C2 (C3×C6).9D12 C6.2PSU3(𝔽2) C33⋊9(C4⋊C4)
C4⋊Dic3 is a maximal quotient of
C12⋊C8 C8⋊Dic3 C24⋊1C4 C24.C4 C6.C42 C4⋊Dic9 Dic3⋊Dic3 C12⋊Dic3 C24.4D6 C30.Q8 C60⋊5C4 C60⋊C4 C42.Q8 C84⋊C4 (C3×C6).9D12 C6.2PSU3(𝔽2) C33⋊9(C4⋊C4)
Matrix representation of C4⋊Dic3 ►in GL3(𝔽13) generated by
| 1 | 0 | 0 |
| 0 | 3 | 6 |
| 0 | 7 | 10 |
| 12 | 0 | 0 |
| 0 | 0 | 12 |
| 0 | 1 | 1 |
| 8 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 8 | 8 |
G:=sub<GL(3,GF(13))| [1,0,0,0,3,7,0,6,10],[12,0,0,0,0,1,0,12,1],[8,0,0,0,5,8,0,0,8] >;
C4⋊Dic3 in GAP, Magma, Sage, TeX
C_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C4:Dic3"); // GroupNames label
G:=SmallGroup(48,13);
// by ID
G=gap.SmallGroup(48,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,20,101,46,804]);
// Polycyclic
G:=Group<a,b,c|a^4=b^6=1,c^2=b^3,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C4⋊Dic3 in TeX
Character table of C4⋊Dic3 in TeX