direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D5, C4⋊1D10, C20⋊C22, D20⋊3C2, C22⋊1D10, D10⋊2C22, C10.5C23, Dic5⋊1C22, C5⋊2(C2×D4), (C4×D5)⋊1C2, (C5×D4)⋊2C2, C5⋊D4⋊1C2, (C2×C10)⋊C22, (C22×D5)⋊2C2, C2.6(C22×D5), SmallGroup(80,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D5
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, C2×C4, D4, D4, C23, D5, D5, C10, C10, C2×D4, Dic5, C20, D10, D10, D10, C2×C10, C4×D5, D20, C5⋊D4, C5×D4, C22×D5, D4×D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22×D5, D4×D5
Character table of D4×D5
| 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);
D4×D5 is a maximal subgroup of
D20⋊C4 D8⋊D5 D40⋊C2 D4⋊6D10 D4⋊8D10 C20⋊D6 D10⋊D6 C20⋊D10 D10⋊D10
D4×D5 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 D4×D5 ►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] >;
D4×D5 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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