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## G = D4×D5order 80 = 24·5

### Direct product of D4 and D5

Aliases: D4×D5, C41D10, C20⋊C22, D203C2, C221D10, D102C22, C10.5C23, Dic51C22, C52(C2×D4), (C4×D5)⋊1C2, (C5×D4)⋊2C2, C5⋊D41C2, (C2×C10)⋊C22, (C22×D5)⋊2C2, C2.6(C22×D5), SmallGroup(80,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4×D5
 Chief series C1 — C5 — C10 — D10 — C22×D5 — D4×D5
 Lower central C5 — C10 — D4×D5
 Upper central C1 — C2 — D4

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

Permutation representations of D4×D5
On 20 points - transitive group 20T21
Generators in S20
(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);

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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