metabelian, supersoluble, monomial
Aliases: D26⋊2C6, Dic13⋊C6, C13⋊D4⋊C3, C13⋊C3⋊2D4, C26.C6⋊C2, C13⋊2(C3×D4), (C2×C26)⋊3C6, C26.5(C2×C6), C22⋊2(C13⋊C6), (C2×C13⋊C6)⋊2C2, C2.5(C2×C13⋊C6), (C22×C13⋊C3)⋊1C2, (C2×C13⋊C3).5C22, SmallGroup(312,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D26⋊C6
G = < a,b,c | a26=b2=c6=1, bab=a-1, cac-1=a3, cbc-1=a15b >
Character table of D26⋊C6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 12A | 12B | 13A | 13B | 26A | 26B | 26C | 26D | 26E | 26F | |
| size | 1 | 1 | 2 | 26 | 13 | 13 | 26 | 13 | 13 | 26 | 26 | 26 | 26 | 26 | 26 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | ζ32 | ζ3 | -1 | ζ32 | ζ3 | ζ6 | ζ32 | ζ3 | ζ65 | ζ65 | ζ6 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 1 | 1 | -1 | 1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ65 | ζ3 | ζ32 | ζ6 | ζ6 | ζ65 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ8 | 1 | 1 | 1 | -1 | ζ32 | ζ3 | -1 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ10 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | ζ32 | ζ3 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | -1 | ζ3 | ζ32 | -1 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ32 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ12 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | ζ3 | ζ32 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | orthogonal lifted from D4 |
| ρ14 | 2 | -2 | 0 | 0 | -1-√-3 | -1+√-3 | 0 | 1+√-3 | 1-√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | complex lifted from C3×D4 |
| ρ15 | 2 | -2 | 0 | 0 | -1+√-3 | -1-√-3 | 0 | 1-√-3 | 1+√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | -2 | 0 | 0 | 0 | -2 | complex lifted from C3×D4 |
| ρ16 | 6 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | orthogonal lifted from C13⋊C6 |
| ρ17 | 6 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -1+√13/2 | -1-√13/2 | -1-√13/2 | -1+√13/2 | -1-√13/2 | -1+√13/2 | orthogonal lifted from C13⋊C6 |
| ρ18 | 6 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | 1+√13/2 | -1+√13/2 | 1-√13/2 | 1+√13/2 | 1-√13/2 | -1-√13/2 | orthogonal lifted from C2×C13⋊C6 |
| ρ19 | 6 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | 1-√13/2 | -1-√13/2 | 1+√13/2 | 1-√13/2 | 1+√13/2 | -1+√13/2 | orthogonal lifted from C2×C13⋊C6 |
| ρ20 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | -ζ1311-ζ138-ζ137+ζ136+ζ135+ζ132 | 1-√13/2 | ζ1312+ζ1310-ζ139+ζ134-ζ133-ζ13 | ζ1311+ζ138+ζ137-ζ136-ζ135-ζ132 | -ζ1312-ζ1310+ζ139-ζ134+ζ133+ζ13 | 1+√13/2 | complex faithful |
| ρ21 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | -ζ1312-ζ1310+ζ139-ζ134+ζ133+ζ13 | 1+√13/2 | -ζ1311-ζ138-ζ137+ζ136+ζ135+ζ132 | ζ1312+ζ1310-ζ139+ζ134-ζ133-ζ13 | ζ1311+ζ138+ζ137-ζ136-ζ135-ζ132 | 1-√13/2 | complex faithful |
| ρ22 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√13/2 | -1-√13/2 | ζ1311+ζ138+ζ137-ζ136-ζ135-ζ132 | 1-√13/2 | -ζ1312-ζ1310+ζ139-ζ134+ζ133+ζ13 | -ζ1311-ζ138-ζ137+ζ136+ζ135+ζ132 | ζ1312+ζ1310-ζ139+ζ134-ζ133-ζ13 | 1+√13/2 | complex faithful |
| ρ23 | 6 | -6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√13/2 | -1+√13/2 | ζ1312+ζ1310-ζ139+ζ134-ζ133-ζ13 | 1+√13/2 | ζ1311+ζ138+ζ137-ζ136-ζ135-ζ132 | -ζ1312-ζ1310+ζ139-ζ134+ζ133+ζ13 | -ζ1311-ζ138-ζ137+ζ136+ζ135+ζ132 | 1-√13/2 | complex faithful |
►On 52 points
Generators in S52
(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 49 50 51 52)
(1 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 52)(23 51)(24 50)(25 49)(26 48)
(2 10 4)(3 19 7)(5 11 13)(6 20 16)(8 12 22)(9 21 25)(15 23 17)(18 24 26)(27 28 37 40 41 50)(29 46 43 42 33 30)(31 38 49 44 51 36)(32 47 52 45 34 39)(35 48)
G:=sub<Sym(52)| (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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,52)(23,51)(24,50)(25,49)(26,48), (2,10,4)(3,19,7)(5,11,13)(6,20,16)(8,12,22)(9,21,25)(15,23,17)(18,24,26)(27,28,37,40,41,50)(29,46,43,42,33,30)(31,38,49,44,51,36)(32,47,52,45,34,39)(35,48)>;
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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52), (1,47)(2,46)(3,45)(4,44)(5,43)(6,42)(7,41)(8,40)(9,39)(10,38)(11,37)(12,36)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,52)(23,51)(24,50)(25,49)(26,48), (2,10,4)(3,19,7)(5,11,13)(6,20,16)(8,12,22)(9,21,25)(15,23,17)(18,24,26)(27,28,37,40,41,50)(29,46,43,42,33,30)(31,38,49,44,51,36)(32,47,52,45,34,39)(35,48) );
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),(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,42),(7,41),(8,40),(9,39),(10,38),(11,37),(12,36),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,52),(23,51),(24,50),(25,49),(26,48)], [(2,10,4),(3,19,7),(5,11,13),(6,20,16),(8,12,22),(9,21,25),(15,23,17),(18,24,26),(27,28,37,40,41,50),(29,46,43,42,33,30),(31,38,49,44,51,36),(32,47,52,45,34,39),(35,48)]])
Matrix representation of D26⋊C6 ►in GL6(𝔽3)
| 2 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 2 | 0 |
| 0 | 1 | 1 | 0 | 2 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 2 |
| 0 | 2 | 1 | 0 | 0 | 0 |
| 0 | 1 | 2 | 0 | 1 | 0 |
| 1 | 0 | 0 | 2 | 0 | 1 |
| 0 | 0 | 2 | 0 | 2 | 0 |
| 1 | 0 | 0 | 1 | 0 | 2 |
G:=sub<GL(6,GF(3))| [2,0,0,1,0,0,0,0,1,0,0,0,0,1,1,0,1,0,0,0,0,1,0,2,0,2,2,0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2,0,0,0,0,1,0,0,0],[2,0,0,1,0,1,0,2,1,0,0,0,0,1,2,0,2,0,0,0,0,2,0,1,0,0,1,0,2,0,2,0,0,1,0,2] >;
D26⋊C6 in GAP, Magma, Sage, TeX
D_{26}\rtimes C_6 % in TeX
G:=Group("D26:C6"); // GroupNames label
G:=SmallGroup(312,12);
// by ID
G=gap.SmallGroup(312,12);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-13,141,7204,464]);
// Polycyclic
G:=Group<a,b,c|a^26=b^2=c^6=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^15*b>;
// generators/relations
Export
Subgroup lattice of D26⋊C6 in TeX
Character table of D26⋊C6 in TeX