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G = D4×D5order 80 = 24·5

Direct product of D4 and D5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C10 — D4×D5
C1C5C10D10C22×D5 — D4×D5
C5C10 — D4×D5
C1C2D4

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 [×6], C4, C4, C22 [×2], C22 [×7], C5, C2×C4, D4, D4 [×3], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C2×D4, Dic5, C20, D10, D10 [×2], D10 [×4], C2×C10 [×2], C4×D5, D20, C5⋊D4 [×2], C5×D4, C22×D5 [×2], D4×D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, D10 [×3], C22×D5, D4×D5

Character table of D4×D5

 class 12A2B2C2D2E2F2G4A4B5A5B10A10B10C10D10E10F20A20B
 size 11225510102102222444444
ρ111111111111111111111    trivial
ρ21111-1-1-1-11-11111111111    linear of order 2
ρ311-1-111-1-1111111-1-1-1-111    linear of order 2
ρ411-1-1-1-1111-11111-1-1-1-111    linear of order 2
ρ5111-1-1-1-11-111111-111-1-1-1    linear of order 2
ρ6111-1111-1-1-11111-111-1-1-1    linear of order 2
ρ711-11-1-11-1-1111111-1-11-1-1    linear of order 2
ρ811-1111-11-1-111111-1-11-1-1    linear of order 2
ρ92-2002-2000022-2-2000000    orthogonal lifted from D4
ρ102-200-22000022-2-2000000    orthogonal lifted from D4
ρ1122-2-2000020-1-5/2-1+5/2-1+5/2-1-5/21-5/21+5/21-5/21+5/2-1+5/2-1-5/2    orthogonal lifted from D10
ρ1222-2-2000020-1+5/2-1-5/2-1-5/2-1+5/21+5/21-5/21+5/21-5/2-1-5/2-1+5/2    orthogonal lifted from D10
ρ132222000020-1+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
ρ142222000020-1-5/2-1+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
ρ1522-220000-20-1+5/2-1-5/2-1-5/2-1+5/2-1-5/21-5/21+5/2-1+5/21+5/21-5/2    orthogonal lifted from D10
ρ16222-20000-20-1+5/2-1-5/2-1-5/2-1+5/21+5/2-1+5/2-1-5/21-5/21+5/21-5/2    orthogonal lifted from D10
ρ17222-20000-20-1-5/2-1+5/2-1+5/2-1-5/21-5/2-1-5/2-1+5/21+5/21-5/21+5/2    orthogonal lifted from D10
ρ1822-220000-20-1-5/2-1+5/2-1+5/2-1-5/2-1+5/21+5/21-5/2-1-5/21-5/21+5/2    orthogonal lifted from D10
ρ194-400000000-1+5-1-51+51-5000000    orthogonal faithful
ρ204-400000000-1-5-1+51-51+5000000    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);

D4×D5 is a maximal subgroup of
D20⋊C4  D8⋊D5  D40⋊C2  D46D10  D48D10  C20⋊D6  D10⋊D6  C20⋊D10  D10⋊D10
D4×D5 is a maximal quotient of
Dic5.14D4  Dic54D4  C22⋊D20  D10.12D4  D10⋊D4  Dic5.5D4  C20⋊Q8  D208C4  D10.13D4  C4⋊D20  D10⋊Q8  D8⋊D5  D83D5  D40⋊C2  SD16⋊D5  SD163D5  Q16⋊D5  Q8.D10  C23⋊D10  C202D4  Dic5⋊D4  C20⋊D4  C20⋊D6  D10⋊D6  C20⋊D10  D10⋊D10

Matrix representation of D4×D5 in GL4(𝔽41) generated by

1000
0100
00040
0010
,
40000
04000
0010
00040
,
6100
40000
0010
0001
,
1600
04000
00400
00040
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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Character table of D4×D5 in TeX

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