metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D5, C5⋊2D8, D20⋊2C2, C4.1D10, C10.7D4, C20.1C22, C5⋊2C8⋊1C2, (C5×D4)⋊1C2, C2.4(C5⋊D4), SmallGroup(80,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D5
G = < a,b,c,d | a4=b2=c5=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Character table of D4⋊D5
| class | 1 | 2A | 2B | 2C | 4 | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | |
| size | 1 | 1 | 4 | 20 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | 0 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ7 | 2 | 2 | -2 | 0 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ8 | 2 | 2 | 2 | 0 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ9 | 2 | 2 | -2 | 0 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ10 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -√2 | √2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ11 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | √2 | -√2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D8 |
| ρ12 | 2 | 2 | 0 | 0 | -2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | ζ53-ζ52 | -ζ54+ζ5 | -ζ53+ζ52 | ζ54-ζ5 | 1+√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
| ρ13 | 2 | 2 | 0 | 0 | -2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -ζ54+ζ5 | -ζ53+ζ52 | ζ54-ζ5 | ζ53-ζ52 | 1-√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ14 | 2 | 2 | 0 | 0 | -2 | -1+√5/2 | -1-√5/2 | 0 | 0 | -1+√5/2 | -1-√5/2 | ζ54-ζ5 | ζ53-ζ52 | -ζ54+ζ5 | -ζ53+ζ52 | 1-√5/2 | 1+√5/2 | complex lifted from C5⋊D4 |
| ρ15 | 2 | 2 | 0 | 0 | -2 | -1-√5/2 | -1+√5/2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -ζ53+ζ52 | ζ54-ζ5 | ζ53-ζ52 | -ζ54+ζ5 | 1+√5/2 | 1-√5/2 | complex lifted from C5⋊D4 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
| ρ17 | 4 | -4 | 0 | 0 | 0 | -1-√5 | -1+√5 | 0 | 0 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful, Schur index 2 |
►On 40 points
Generators in S40
(1 19 9 14)(2 20 10 15)(3 16 6 11)(4 17 7 12)(5 18 8 13)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 39)(2 40)(3 36)(4 37)(5 38)(6 31)(7 32)(8 33)(9 34)(10 35)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)
(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)
(1 5)(2 4)(7 10)(8 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)
G:=sub<Sym(40)| (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (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), (1,5)(2,4)(7,10)(8,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37)>;
G:=Group( (1,19,9,14)(2,20,10,15)(3,16,6,11)(4,17,7,12)(5,18,8,13)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40), (1,39)(2,40)(3,36)(4,37)(5,38)(6,31)(7,32)(8,33)(9,34)(10,35)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30), (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), (1,5)(2,4)(7,10)(8,9)(11,16)(12,20)(13,19)(14,18)(15,17)(21,31)(22,35)(23,34)(24,33)(25,32)(26,36)(27,40)(28,39)(29,38)(30,37) );
G=PermutationGroup([[(1,19,9,14),(2,20,10,15),(3,16,6,11),(4,17,7,12),(5,18,8,13),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40)], [(1,39),(2,40),(3,36),(4,37),(5,38),(6,31),(7,32),(8,33),(9,34),(10,35),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30)], [(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)], [(1,5),(2,4),(7,10),(8,9),(11,16),(12,20),(13,19),(14,18),(15,17),(21,31),(22,35),(23,34),(24,33),(25,32),(26,36),(27,40),(28,39),(29,38),(30,37)]])
D4⋊D5 is a maximal subgroup of
D5×D8 D8⋊D5 D40⋊C2 SD16⋊3D5 D4.D10 D4⋊D10 D4.8D10 C15⋊D8 C5⋊D24 D4⋊D15 D4⋊D25 C52⋊2D8 C5⋊D40 C52⋊7D8
D4⋊D5 is a maximal quotient of
C10.D8 D20⋊6C4 C5⋊D16 D8.D5 C5⋊SD32 C5⋊Q32 D4⋊Dic5 C15⋊D8 C5⋊D24 D4⋊D15 D4⋊D25 C52⋊2D8 C5⋊D40 C52⋊7D8
Matrix representation of D4⋊D5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 24 |
| 0 | 0 | 12 | 0 |
| 34 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 34 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,39,40],[1,0,0,0,0,1,0,0,0,0,0,12,0,0,24,0],[34,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,34,40,0,0,0,0,1,1,0,0,0,40] >;
D4⋊D5 in GAP, Magma, Sage, TeX
D_4\rtimes D_5
% in TeX
G:=Group("D4:D5"); // GroupNames label
G:=SmallGroup(80,15);
// by ID
G=gap.SmallGroup(80,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,61,182,97,42,1604]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^5=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of D4⋊D5 in TeX
Character table of D4⋊D5 in TeX