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G = D5×D9order 180 = 22·32·5

Direct product of D5 and D9

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

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

C1C45 — D5×D9
C1C3C15C45C9×D5 — D5×D9
C45 — D5×D9
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 >

5C2
9C2
45C2
45C22
3S3
5C6
15S3
9C10
9D5
15D6
5C18
5D9
9D10
3C5×S3
3D15
5D18
3S3×D5

Character table of D5×D9

 class 12A2B2C35A5B69A9B9C10A10B15A15B18A18B18C45A45B45C45D45E45F
 size 1594522210222181844101010444444
ρ1111111111111111111111111    trivial
ρ21-11-1111-11111111-1-1-1111111    linear of order 2
ρ311-1-11111111-1-111111111111    linear of order 2
ρ41-1-11111-1111-1-111-1-1-1111111    linear of order 2
ρ522002222-1-1-10022-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62-200222-2-1-1-10022111-1-1-1-1-1-1    orthogonal lifted from D6
ρ720-202-1-5/2-1+5/202221-5/21+5/2-1+5/2-1-5/2000-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D10
ρ820202-1-5/2-1+5/20222-1+5/2-1-5/2-1+5/2-1-5/2000-1+5/2-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ920202-1+5/2-1-5/20222-1-5/2-1+5/2-1-5/2-1+5/2000-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ1020-202-1+5/2-1-5/202221+5/21-5/2-1-5/2-1+5/2000-1-5/2-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2    orthogonal lifted from D10
ρ112-200-1221ζ9792ζ989ζ959400-1-197929594989ζ9594ζ989ζ989ζ9792ζ9594ζ9792    orthogonal lifted from D18
ρ122-200-1221ζ989ζ9594ζ979200-1-198997929594ζ9792ζ9594ζ9594ζ989ζ9792ζ989    orthogonal lifted from D18
ρ132-200-1221ζ9594ζ9792ζ98900-1-195949899792ζ989ζ9792ζ9792ζ9594ζ989ζ9594    orthogonal lifted from D18
ρ142200-122-1ζ989ζ9594ζ979200-1-1ζ989ζ9792ζ9594ζ9792ζ9594ζ9594ζ989ζ9792ζ989    orthogonal lifted from D9
ρ152200-122-1ζ9594ζ9792ζ98900-1-1ζ9594ζ989ζ9792ζ989ζ9792ζ9792ζ9594ζ989ζ9594    orthogonal lifted from D9
ρ162200-122-1ζ9792ζ989ζ959400-1-1ζ9792ζ9594ζ989ζ9594ζ989ζ989ζ9792ζ9594ζ9792    orthogonal lifted from D9
ρ1740004-1-5-1+50-2-2-200-1+5-1-50001-5/21+5/21-5/21-5/21+5/21+5/2    orthogonal lifted from S3×D5
ρ1840004-1+5-1-50-2-2-200-1-5-1+50001+5/21-5/21+5/21+5/21-5/21-5/2    orthogonal lifted from S3×D5
ρ194000-2-1-5-1+5097+2ζ9298+2ζ995+2ζ94001-5/21+5/2000ζ95ζ5495ζ594ζ5494ζ5ζ98ζ5398ζ529ζ539ζ52ζ98ζ5498ζ59ζ549ζ5ζ97ζ5497ζ592ζ5492ζ5ζ95ζ5395ζ5294ζ5394ζ52ζ97ζ5397ζ5292ζ5392ζ52    orthogonal faithful
ρ204000-2-1+5-1-5097+2ζ9298+2ζ995+2ζ94001+5/21-5/2000ζ95ζ5395ζ5294ζ5394ζ52ζ98ζ5498ζ59ζ549ζ5ζ98ζ5398ζ529ζ539ζ52ζ97ζ5397ζ5292ζ5392ζ52ζ95ζ5495ζ594ζ5494ζ5ζ97ζ5497ζ592ζ5492ζ5    orthogonal faithful
ρ214000-2-1+5-1-5098+2ζ995+2ζ9497+2ζ92001+5/21-5/2000ζ97ζ5397ζ5292ζ5392ζ52ζ95ζ5495ζ594ζ5494ζ5ζ95ζ5395ζ5294ζ5394ζ52ζ98ζ5398ζ529ζ539ζ52ζ97ζ5497ζ592ζ5492ζ5ζ98ζ5498ζ59ζ549ζ5    orthogonal faithful
ρ224000-2-1+5-1-5095+2ζ9497+2ζ9298+2ζ9001+5/21-5/2000ζ98ζ5398ζ529ζ539ζ52ζ97ζ5497ζ592ζ5492ζ5ζ97ζ5397ζ5292ζ5392ζ52ζ95ζ5395ζ5294ζ5394ζ52ζ98ζ5498ζ59ζ549ζ5ζ95ζ5495ζ594ζ5494ζ5    orthogonal faithful
ρ234000-2-1-5-1+5095+2ζ9497+2ζ9298+2ζ9001-5/21+5/2000ζ98ζ5498ζ59ζ549ζ5ζ97ζ5397ζ5292ζ5392ζ52ζ97ζ5497ζ592ζ5492ζ5ζ95ζ5495ζ594ζ5494ζ5ζ98ζ5398ζ529ζ539ζ52ζ95ζ5395ζ5294ζ5394ζ52    orthogonal faithful
ρ244000-2-1-5-1+5098+2ζ995+2ζ9497+2ζ92001-5/21+5/2000ζ97ζ5497ζ592ζ5492ζ5ζ95ζ5395ζ5294ζ5394ζ52ζ95ζ5495ζ594ζ5494ζ5ζ98ζ5498ζ59ζ549ζ5ζ97ζ5397ζ5292ζ5392ζ52ζ98ζ5398ζ529ζ539ζ52    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

1000
0100
001801
0016614
,
1000
0100
001800
001661
,
5017700
45400
0010
0001
,
505400
413100
0010
0001
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

Export

Subgroup lattice of D5×D9 in TeX
Character table of D5×D9 in TeX

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