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## G = D9×F5order 360 = 23·32·5

### Direct product of D9 and F5

Aliases: D9×F5, D45⋊C4, D5.1D18, C5⋊(C4×D9), C9⋊F5⋊C2, C45⋊(C2×C4), (C5×D9)⋊C4, (C9×F5)⋊C2, C91(C2×F5), C3.(S3×F5), C15.(C4×S3), (D5×D9).C2, (C3×F5).S3, (C3×D5).1D6, (C9×D5).C22, SmallGroup(360,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C45 — D9×F5
 Chief series C1 — C3 — C15 — C45 — C9×D5 — C9×F5 — D9×F5
 Lower central C45 — D9×F5
 Upper central C1

Generators and relations for D9×F5
G = < a,b,c,d | a9=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Character table of D9×F5

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5 6 9A 9B 9C 10 12A 12B 15 18A 18B 18C 36A 36B 36C 36D 36E 36F 45A 45B 45C size 1 5 9 45 2 5 5 45 45 4 10 2 2 2 36 10 10 8 10 10 10 10 10 10 10 10 10 8 8 8 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 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 -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 -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 1 1 1 1 1 1 linear of order 2 ρ5 1 -1 1 -1 1 -i i -i i 1 -1 1 1 1 1 -i i 1 -1 -1 -1 i -i -i -i i i 1 1 1 linear of order 4 ρ6 1 -1 -1 1 1 -i i i -i 1 -1 1 1 1 -1 -i i 1 -1 -1 -1 i -i -i -i i i 1 1 1 linear of order 4 ρ7 1 -1 -1 1 1 i -i -i i 1 -1 1 1 1 -1 i -i 1 -1 -1 -1 -i i i i -i -i 1 1 1 linear of order 4 ρ8 1 -1 1 -1 1 i -i i -i 1 -1 1 1 1 1 i -i 1 -1 -1 -1 -i i i i -i -i 1 1 1 linear of order 4 ρ9 2 2 0 0 2 -2 -2 0 0 2 2 -1 -1 -1 0 -2 -2 2 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 orthogonal lifted from D6 ρ10 2 2 0 0 2 2 2 0 0 2 2 -1 -1 -1 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 0 0 -1 2 2 0 0 2 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D9 ρ12 2 2 0 0 -1 -2 -2 0 0 2 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 1 1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D18 ρ13 2 2 0 0 -1 -2 -2 0 0 2 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 1 1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D18 ρ14 2 2 0 0 -1 2 2 0 0 2 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 orthogonal lifted from D9 ρ15 2 2 0 0 -1 -2 -2 0 0 2 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 1 1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 orthogonal lifted from D18 ρ16 2 2 0 0 -1 2 2 0 0 2 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 orthogonal lifted from D9 ρ17 2 -2 0 0 2 -2i 2i 0 0 2 -2 -1 -1 -1 0 -2i 2i 2 1 1 1 -i i i i -i -i -1 -1 -1 complex lifted from C4×S3 ρ18 2 -2 0 0 2 2i -2i 0 0 2 -2 -1 -1 -1 0 2i -2i 2 1 1 1 i -i -i -i i i -1 -1 -1 complex lifted from C4×S3 ρ19 2 -2 0 0 -1 -2i 2i 0 0 2 1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 i -i -1 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ4ζ98+ζ4ζ9 ζ43ζ97+ζ43ζ92 ζ43ζ95+ζ43ζ94 ζ43ζ98+ζ43ζ9 ζ4ζ97+ζ4ζ92 ζ4ζ95+ζ4ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 complex lifted from C4×D9 ρ20 2 -2 0 0 -1 2i -2i 0 0 2 1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 -i i -1 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ43ζ97+ζ43ζ92 ζ4ζ95+ζ4ζ94 ζ4ζ98+ζ4ζ9 ζ4ζ97+ζ4ζ92 ζ43ζ95+ζ43ζ94 ζ43ζ98+ζ43ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 complex lifted from C4×D9 ρ21 2 -2 0 0 -1 2i -2i 0 0 2 1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 -i i -1 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ43ζ95+ζ43ζ94 ζ4ζ98+ζ4ζ9 ζ4ζ97+ζ4ζ92 ζ4ζ95+ζ4ζ94 ζ43ζ98+ζ43ζ9 ζ43ζ97+ζ43ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 complex lifted from C4×D9 ρ22 2 -2 0 0 -1 2i -2i 0 0 2 1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 -i i -1 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 ζ43ζ98+ζ43ζ9 ζ4ζ97+ζ4ζ92 ζ4ζ95+ζ4ζ94 ζ4ζ98+ζ4ζ9 ζ43ζ97+ζ43ζ92 ζ43ζ95+ζ43ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 complex lifted from C4×D9 ρ23 2 -2 0 0 -1 -2i 2i 0 0 2 1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 i -i -1 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 ζ4ζ97+ζ4ζ92 ζ43ζ95+ζ43ζ94 ζ43ζ98+ζ43ζ9 ζ43ζ97+ζ43ζ92 ζ4ζ95+ζ4ζ94 ζ4ζ98+ζ4ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 complex lifted from C4×D9 ρ24 2 -2 0 0 -1 -2i 2i 0 0 2 1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 i -i -1 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 ζ4ζ95+ζ4ζ94 ζ43ζ98+ζ43ζ9 ζ43ζ97+ζ43ζ92 ζ43ζ95+ζ43ζ94 ζ4ζ98+ζ4ζ9 ζ4ζ97+ζ4ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 complex lifted from C4×D9 ρ25 4 0 -4 0 4 0 0 0 0 -1 0 4 4 4 1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from C2×F5 ρ26 4 0 4 0 4 0 0 0 0 -1 0 4 4 4 -1 0 0 -1 0 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from F5 ρ27 8 0 0 0 8 0 0 0 0 -2 0 -4 -4 -4 0 0 0 -2 0 0 0 0 0 0 0 0 0 1 1 1 orthogonal lifted from S3×F5 ρ28 8 0 0 0 -4 0 0 0 0 -2 0 4ζ98+4ζ9 4ζ97+4ζ92 4ζ95+4ζ94 0 0 0 1 0 0 0 0 0 0 0 0 0 -ζ97-ζ92 -ζ98-ζ9 -ζ95-ζ94 orthogonal faithful ρ29 8 0 0 0 -4 0 0 0 0 -2 0 4ζ97+4ζ92 4ζ95+4ζ94 4ζ98+4ζ9 0 0 0 1 0 0 0 0 0 0 0 0 0 -ζ95-ζ94 -ζ97-ζ92 -ζ98-ζ9 orthogonal faithful ρ30 8 0 0 0 -4 0 0 0 0 -2 0 4ζ95+4ζ94 4ζ98+4ζ9 4ζ97+4ζ92 0 0 0 1 0 0 0 0 0 0 0 0 0 -ζ98-ζ9 -ζ95-ζ94 -ζ97-ζ92 orthogonal faithful

Smallest permutation representation of D9×F5
On 45 points
Generators in S45
(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 15)(11 14)(12 13)(16 18)(19 26)(20 25)(21 24)(22 23)(28 36)(29 35)(30 34)(31 33)(37 42)(38 41)(39 40)(43 45)
(1 13 23 40 28)(2 14 24 41 29)(3 15 25 42 30)(4 16 26 43 31)(5 17 27 44 32)(6 18 19 45 33)(7 10 20 37 34)(8 11 21 38 35)(9 12 22 39 36)
(10 20 34 37)(11 21 35 38)(12 22 36 39)(13 23 28 40)(14 24 29 41)(15 25 30 42)(16 26 31 43)(17 27 32 44)(18 19 33 45)

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

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

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

Matrix representation of D9×F5 in GL6(𝔽181)

 131 54 0 0 0 0 127 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 177 54 0 0 0 0 50 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 180 0 0 1 0 0 180 0 0 0 1 0 180 0 0 0 0 1 180
,
 19 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0

G:=sub<GL(6,GF(181))| [131,127,0,0,0,0,54,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[177,50,0,0,0,0,54,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,180,180,180,180],[19,0,0,0,0,0,0,19,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;

D9×F5 in GAP, Magma, Sage, TeX

D_9\times F_5
% in TeX

G:=Group("D9xF5");
// GroupNames label

G:=SmallGroup(360,39);
// by ID

G=gap.SmallGroup(360,39);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-5,-3,24,1641,741,1444,736,4331]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^5=d^4=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^-1=c^3>;
// generators/relations

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