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## G = D5×D9order 180 = 22·32·5

### Direct product of D5 and D9

Aliases: D5×D9, D45⋊C2, C91D10, C51D18, C45⋊C22, C15.D6, (C5×D9)⋊C2, (C9×D5)⋊C2, C3.(S3×D5), (C3×D5).1S3, SmallGroup(180,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C45 — D5×D9
 Chief series C1 — C3 — C15 — C45 — C9×D5 — D5×D9
 Lower central C45 — D5×D9
 Upper central C1

Generators and relations for D5×D9
G = < a,b,c,d | a5=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of D5×D9

 class 1 2A 2B 2C 3 5A 5B 6 9A 9B 9C 10A 10B 15A 15B 18A 18B 18C 45A 45B 45C 45D 45E 45F size 1 5 9 45 2 2 2 10 2 2 2 18 18 4 4 10 10 10 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 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 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 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 1 1 1 1 linear of order 2 ρ5 2 2 0 0 2 2 2 2 -1 -1 -1 0 0 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 -2 0 0 2 2 2 -2 -1 -1 -1 0 0 2 2 1 1 1 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ7 2 0 -2 0 2 -1-√5/2 -1+√5/2 0 2 2 2 1-√5/2 1+√5/2 -1+√5/2 -1-√5/2 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D10 ρ8 2 0 2 0 2 -1-√5/2 -1+√5/2 0 2 2 2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 0 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 0 2 0 2 -1+√5/2 -1-√5/2 0 2 2 2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ10 2 0 -2 0 2 -1+√5/2 -1-√5/2 0 2 2 2 1+√5/2 1-√5/2 -1-√5/2 -1+√5/2 0 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D10 ρ11 2 -2 0 0 -1 2 2 1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 0 0 -1 -1 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D18 ρ12 2 -2 0 0 -1 2 2 1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 0 0 -1 -1 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D18 ρ13 2 -2 0 0 -1 2 2 1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 0 0 -1 -1 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D18 ρ14 2 2 0 0 -1 2 2 -1 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 0 0 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ15 2 2 0 0 -1 2 2 -1 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 0 0 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ16 2 2 0 0 -1 2 2 -1 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 0 0 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ17 4 0 0 0 4 -1-√5 -1+√5 0 -2 -2 -2 0 0 -1+√5 -1-√5 0 0 0 1-√5/2 1+√5/2 1-√5/2 1-√5/2 1+√5/2 1+√5/2 orthogonal lifted from S3×D5 ρ18 4 0 0 0 4 -1+√5 -1-√5 0 -2 -2 -2 0 0 -1-√5 -1+√5 0 0 0 1+√5/2 1-√5/2 1+√5/2 1+√5/2 1-√5/2 1-√5/2 orthogonal lifted from S3×D5 ρ19 4 0 0 0 -2 -1-√5 -1+√5 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 0 0 1-√5/2 1+√5/2 0 0 0 ζ95ζ54+ζ95ζ5+ζ94ζ54+ζ94ζ5 ζ98ζ53+ζ98ζ52+ζ9ζ53+ζ9ζ52 ζ98ζ54+ζ98ζ5+ζ9ζ54+ζ9ζ5 ζ97ζ54+ζ97ζ5+ζ92ζ54+ζ92ζ5 ζ95ζ53+ζ95ζ52+ζ94ζ53+ζ94ζ52 ζ97ζ53+ζ97ζ52+ζ92ζ53+ζ92ζ52 orthogonal faithful ρ20 4 0 0 0 -2 -1+√5 -1-√5 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 0 0 1+√5/2 1-√5/2 0 0 0 ζ95ζ53+ζ95ζ52+ζ94ζ53+ζ94ζ52 ζ98ζ54+ζ98ζ5+ζ9ζ54+ζ9ζ5 ζ98ζ53+ζ98ζ52+ζ9ζ53+ζ9ζ52 ζ97ζ53+ζ97ζ52+ζ92ζ53+ζ92ζ52 ζ95ζ54+ζ95ζ5+ζ94ζ54+ζ94ζ5 ζ97ζ54+ζ97ζ5+ζ92ζ54+ζ92ζ5 orthogonal faithful ρ21 4 0 0 0 -2 -1+√5 -1-√5 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 0 0 1+√5/2 1-√5/2 0 0 0 ζ97ζ53+ζ97ζ52+ζ92ζ53+ζ92ζ52 ζ95ζ54+ζ95ζ5+ζ94ζ54+ζ94ζ5 ζ95ζ53+ζ95ζ52+ζ94ζ53+ζ94ζ52 ζ98ζ53+ζ98ζ52+ζ9ζ53+ζ9ζ52 ζ97ζ54+ζ97ζ5+ζ92ζ54+ζ92ζ5 ζ98ζ54+ζ98ζ5+ζ9ζ54+ζ9ζ5 orthogonal faithful ρ22 4 0 0 0 -2 -1+√5 -1-√5 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 0 0 1+√5/2 1-√5/2 0 0 0 ζ98ζ53+ζ98ζ52+ζ9ζ53+ζ9ζ52 ζ97ζ54+ζ97ζ5+ζ92ζ54+ζ92ζ5 ζ97ζ53+ζ97ζ52+ζ92ζ53+ζ92ζ52 ζ95ζ53+ζ95ζ52+ζ94ζ53+ζ94ζ52 ζ98ζ54+ζ98ζ5+ζ9ζ54+ζ9ζ5 ζ95ζ54+ζ95ζ5+ζ94ζ54+ζ94ζ5 orthogonal faithful ρ23 4 0 0 0 -2 -1-√5 -1+√5 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 0 0 1-√5/2 1+√5/2 0 0 0 ζ98ζ54+ζ98ζ5+ζ9ζ54+ζ9ζ5 ζ97ζ53+ζ97ζ52+ζ92ζ53+ζ92ζ52 ζ97ζ54+ζ97ζ5+ζ92ζ54+ζ92ζ5 ζ95ζ54+ζ95ζ5+ζ94ζ54+ζ94ζ5 ζ98ζ53+ζ98ζ52+ζ9ζ53+ζ9ζ52 ζ95ζ53+ζ95ζ52+ζ94ζ53+ζ94ζ52 orthogonal faithful ρ24 4 0 0 0 -2 -1-√5 -1+√5 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 0 0 1-√5/2 1+√5/2 0 0 0 ζ97ζ54+ζ97ζ5+ζ92ζ54+ζ92ζ5 ζ95ζ53+ζ95ζ52+ζ94ζ53+ζ94ζ52 ζ95ζ54+ζ95ζ5+ζ94ζ54+ζ94ζ5 ζ98ζ54+ζ98ζ5+ζ9ζ54+ζ9ζ5 ζ97ζ53+ζ97ζ52+ζ92ζ53+ζ92ζ52 ζ98ζ53+ζ98ζ52+ζ9ζ53+ζ9ζ52 orthogonal faithful

Smallest permutation representation of D5×D9
On 45 points
Generators in S45
(1 28 40 23 14)(2 29 41 24 15)(3 30 42 25 16)(4 31 43 26 17)(5 32 44 27 18)(6 33 45 19 10)(7 34 37 20 11)(8 35 38 21 12)(9 36 39 22 13)
(1 14)(2 15)(3 16)(4 17)(5 18)(6 10)(7 11)(8 12)(9 13)(19 33)(20 34)(21 35)(22 36)(23 28)(24 29)(25 30)(26 31)(27 32)
(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 41 42 43 44 45)
(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 26)(20 25)(21 24)(22 23)(28 36)(29 35)(30 34)(31 33)(37 42)(38 41)(39 40)(43 45)

G:=sub<Sym(45)| (1,28,40,23,14)(2,29,41,24,15)(3,30,42,25,16)(4,31,43,26,17)(5,32,44,27,18)(6,33,45,19,10)(7,34,37,20,11)(8,35,38,21,12)(9,36,39,22,13), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,33)(20,34)(21,35)(22,36)(23,28)(24,29)(25,30)(26,31)(27,32), (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)(37,38,39,40,41,42,43,44,45), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,36)(29,35)(30,34)(31,33)(37,42)(38,41)(39,40)(43,45)>;

G:=Group( (1,28,40,23,14)(2,29,41,24,15)(3,30,42,25,16)(4,31,43,26,17)(5,32,44,27,18)(6,33,45,19,10)(7,34,37,20,11)(8,35,38,21,12)(9,36,39,22,13), (1,14)(2,15)(3,16)(4,17)(5,18)(6,10)(7,11)(8,12)(9,13)(19,33)(20,34)(21,35)(22,36)(23,28)(24,29)(25,30)(26,31)(27,32), (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)(37,38,39,40,41,42,43,44,45), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,26)(20,25)(21,24)(22,23)(28,36)(29,35)(30,34)(31,33)(37,42)(38,41)(39,40)(43,45) );

G=PermutationGroup([(1,28,40,23,14),(2,29,41,24,15),(3,30,42,25,16),(4,31,43,26,17),(5,32,44,27,18),(6,33,45,19,10),(7,34,37,20,11),(8,35,38,21,12),(9,36,39,22,13)], [(1,14),(2,15),(3,16),(4,17),(5,18),(6,10),(7,11),(8,12),(9,13),(19,33),(20,34),(21,35),(22,36),(23,28),(24,29),(25,30),(26,31),(27,32)], [(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),(37,38,39,40,41,42,43,44,45)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,26),(20,25),(21,24),(22,23),(28,36),(29,35),(30,34),(31,33),(37,42),(38,41),(39,40),(43,45)])

D5×D9 is a maximal quotient of   C45⋊Q8  D90.C2  C5⋊D36  C45⋊D4  C9⋊D20

Matrix representation of D5×D9 in GL4(𝔽181) generated by

 1 0 0 0 0 1 0 0 0 0 180 1 0 0 166 14
,
 1 0 0 0 0 1 0 0 0 0 180 0 0 0 166 1
,
 50 177 0 0 4 54 0 0 0 0 1 0 0 0 0 1
,
 50 54 0 0 4 131 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(181))| [1,0,0,0,0,1,0,0,0,0,180,166,0,0,1,14],[1,0,0,0,0,1,0,0,0,0,180,166,0,0,0,1],[50,4,0,0,177,54,0,0,0,0,1,0,0,0,0,1],[50,4,0,0,54,131,0,0,0,0,1,0,0,0,0,1] >;

D5×D9 in GAP, Magma, Sage, TeX

D_5\times D_9
% in TeX

G:=Group("D5xD9");
// GroupNames label

G:=SmallGroup(180,7);
// by ID

G=gap.SmallGroup(180,7);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-3,517,462,963,1509]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^9=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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