direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4xD5, C4:1D10, C20:C22, D20:3C2, C22:1D10, D10:2C22, C10.5C23, Dic5:1C22, C5:2(C2xD4), (C4xD5):1C2, (C5xD4):2C2, C5:D4:1C2, (C2xC10):C22, (C22xD5):2C2, C2.6(C22xD5), SmallGroup(80,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4xD5
G = < a,b,c,d | a4=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 170 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2xC4, D4, D4, C23, D5, D5, C10, C10, C2xD4, Dic5, C20, D10, D10, D10, C2xC10, C4xD5, D20, C5:D4, C5xD4, C22xD5, D4xD5
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C22xD5, D4xD5
Character table of D4xD5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 5A | 5B | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | |
| size | 1 | 1 | 2 | 2 | 5 | 5 | 10 | 10 | 2 | 10 | 2 | 2 | 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 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 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D10 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 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 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 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 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 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 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ15 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 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 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ16 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 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 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ17 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 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 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ18 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -2 | 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 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√5 | -1-√5 | 1+√5 | 1-√5 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√5 | -1+√5 | 1-√5 | 1+√5 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 20 points - transitive group 20T21
(1 14 9 19)(2 15 10 20)(3 11 6 16)(4 12 7 17)(5 13 8 18)
(11 16)(12 17)(13 18)(14 19)(15 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)
G:=sub<Sym(20)| (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18), (11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)>;
G:=Group( (1,14,9,19)(2,15,10,20)(3,11,6,16)(4,12,7,17)(5,13,8,18), (11,16)(12,17)(13,18)(14,19)(15,20), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19) );
G=PermutationGroup([[(1,14,9,19),(2,15,10,20),(3,11,6,16),(4,12,7,17),(5,13,8,18)], [(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19)]])
G:=TransitiveGroup(20,21);
D4xD5 is a maximal subgroup of
D20:C4 D8:D5 D40:C2 D4:6D10 D4:8D10 C20:D6 D10:D6 C20:D10 D10:D10
D4xD5 is a maximal quotient of
Dic5.14D4 Dic5:4D4 C22:D20 D10.12D4 D10:D4 Dic5.5D4 C20:Q8 D20:8C4 D10.13D4 C4:D20 D10:Q8 D8:D5 D8:3D5 D40:C2 SD16:D5 SD16:3D5 Q16:D5 Q8.D10 C23:D10 C20:2D4 Dic5:D4 C20:D4 C20:D6 D10:D6 C20:D10 D10:D10
Matrix representation of D4xD5 ►in GL4(F41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 6 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 6 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[6,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,6,40,0,0,0,0,40,0,0,0,0,40] >;
D4xD5 in GAP, Magma, Sage, TeX
D_4\times D_5
% in TeX
G:=Group("D4xD5"); // GroupNames label
G:=SmallGroup(80,39);
// by ID
G=gap.SmallGroup(80,39);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-5,97,1604]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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