metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D5, C5⋊2SD16, C4.2D10, C10.8D4, Dic10⋊2C2, C20.2C22, C5⋊2C8⋊2C2, (C5×D4).1C2, C2.5(C5⋊D4), SmallGroup(80,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D5
G = < a,b,c,d | a4=b2=c5=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Character table of D4.D5
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | |
| size | 1 | 1 | 4 | 2 | 20 | 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 | -2 | 0 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
| ρ6 | 2 | 2 | 2 | 2 | 0 | -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 | 2 | 0 | -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 |
| ρ8 | 2 | 2 | -2 | 2 | 0 | -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 |
| ρ9 | 2 | 2 | -2 | 2 | 0 | -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 | -2 | 0 | -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 |
| ρ11 | 2 | 2 | 0 | -2 | 0 | -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 |
| ρ12 | 2 | 2 | 0 | -2 | 0 | -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 |
| ρ13 | 2 | 2 | 0 | -2 | 0 | -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 |
| ρ14 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | -√-2 | √-2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ15 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | √-2 | -√-2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from SD16 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | -1+√5 | -1-√5 | 0 | 0 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic 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 | symplectic 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 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26)(22 27)(23 28)(24 29)(25 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 30 9 25)(2 29 10 24)(3 28 6 23)(4 27 7 22)(5 26 8 21)(11 38 16 33)(12 37 17 32)(13 36 18 31)(14 40 19 35)(15 39 20 34)
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), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,26)(22,27)(23,28)(24,29)(25,30), (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), (1,30,9,25)(2,29,10,24)(3,28,6,23)(4,27,7,22)(5,26,8,21)(11,38,16,33)(12,37,17,32)(13,36,18,31)(14,40,19,35)(15,39,20,34)>;
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), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,26)(22,27)(23,28)(24,29)(25,30), (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), (1,30,9,25)(2,29,10,24)(3,28,6,23)(4,27,7,22)(5,26,8,21)(11,38,16,33)(12,37,17,32)(13,36,18,31)(14,40,19,35)(15,39,20,34) );
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)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,26),(22,27),(23,28),(24,29),(25,30)], [(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)], [(1,30,9,25),(2,29,10,24),(3,28,6,23),(4,27,7,22),(5,26,8,21),(11,38,16,33),(12,37,17,32),(13,36,18,31),(14,40,19,35),(15,39,20,34)]])
D4.D5 is a maximal subgroup of
D8⋊D5 D8⋊3D5 D5×SD16 SD16⋊D5 D4.D10 D4.8D10 D4.9D10 C20.D6 D12.D5 D4.D15 D4.D25 D20.D5 C52⋊3SD16 C52⋊8SD16
D4.D5 is a maximal quotient of
C20.Q8 C10.Q16 D4⋊Dic5 C20.D6 D12.D5 D4.D15 D4.D25 D20.D5 C52⋊3SD16 C52⋊8SD16
Matrix representation of D4.D5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 37 |
| 0 | 0 | 21 | 1 |
| 40 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 |
| 0 | 0 | 40 | 37 |
| 0 | 0 | 0 | 1 |
| 16 | 0 | 0 | 0 |
| 18 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 28 | 0 | 0 |
| 9 | 35 | 0 | 0 |
| 0 | 0 | 11 | 22 |
| 0 | 0 | 28 | 30 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,21,0,0,37,1],[40,18,0,0,0,1,0,0,0,0,40,0,0,0,37,1],[16,18,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[6,9,0,0,28,35,0,0,0,0,11,28,0,0,22,30] >;
D4.D5 in GAP, Magma, Sage, TeX
D_4.D_5
% in TeX
G:=Group("D4.D5"); // GroupNames label
G:=SmallGroup(80,16);
// by ID
G=gap.SmallGroup(80,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,40,61,182,97,42,1604]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^5=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of D4.D5 in TeX
Character table of D4.D5 in TeX