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## G = C5×D9order 90 = 2·32·5

### Direct product of C5 and D9

Aliases: C5×D9, C9⋊C10, C452C2, C15.2S3, C3.(C5×S3), SmallGroup(90,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C5×D9
 Chief series C1 — C3 — C9 — C45 — C5×D9
 Lower central C9 — C5×D9
 Upper central C1 — C5

Generators and relations for C5×D9
G = < a,b,c | a5=b9=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of C5×D9

 class 1 2 3 5A 5B 5C 5D 9A 9B 9C 10A 10B 10C 10D 15A 15B 15C 15D 45A 45B 45C 45D 45E 45F 45G 45H 45I 45J 45K 45L size 1 9 2 1 1 1 1 2 2 2 9 9 9 9 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 ζ5 ζ52 ζ53 ζ54 1 1 1 -ζ53 -ζ52 -ζ54 -ζ5 ζ52 ζ5 ζ54 ζ53 ζ5 ζ5 ζ54 ζ54 ζ53 ζ53 ζ53 ζ54 ζ52 ζ52 ζ5 ζ52 linear of order 10 ρ4 1 1 1 ζ53 ζ5 ζ54 ζ52 1 1 1 ζ54 ζ5 ζ52 ζ53 ζ5 ζ53 ζ52 ζ54 ζ53 ζ53 ζ52 ζ52 ζ54 ζ54 ζ54 ζ52 ζ5 ζ5 ζ53 ζ5 linear of order 5 ρ5 1 -1 1 ζ52 ζ54 ζ5 ζ53 1 1 1 -ζ5 -ζ54 -ζ53 -ζ52 ζ54 ζ52 ζ53 ζ5 ζ52 ζ52 ζ53 ζ53 ζ5 ζ5 ζ5 ζ53 ζ54 ζ54 ζ52 ζ54 linear of order 10 ρ6 1 1 1 ζ54 ζ53 ζ52 ζ5 1 1 1 ζ52 ζ53 ζ5 ζ54 ζ53 ζ54 ζ5 ζ52 ζ54 ζ54 ζ5 ζ5 ζ52 ζ52 ζ52 ζ5 ζ53 ζ53 ζ54 ζ53 linear of order 5 ρ7 1 -1 1 ζ53 ζ5 ζ54 ζ52 1 1 1 -ζ54 -ζ5 -ζ52 -ζ53 ζ5 ζ53 ζ52 ζ54 ζ53 ζ53 ζ52 ζ52 ζ54 ζ54 ζ54 ζ52 ζ5 ζ5 ζ53 ζ5 linear of order 10 ρ8 1 1 1 ζ5 ζ52 ζ53 ζ54 1 1 1 ζ53 ζ52 ζ54 ζ5 ζ52 ζ5 ζ54 ζ53 ζ5 ζ5 ζ54 ζ54 ζ53 ζ53 ζ53 ζ54 ζ52 ζ52 ζ5 ζ52 linear of order 5 ρ9 1 -1 1 ζ54 ζ53 ζ52 ζ5 1 1 1 -ζ52 -ζ53 -ζ5 -ζ54 ζ53 ζ54 ζ5 ζ52 ζ54 ζ54 ζ5 ζ5 ζ52 ζ52 ζ52 ζ5 ζ53 ζ53 ζ54 ζ53 linear of order 10 ρ10 1 1 1 ζ52 ζ54 ζ5 ζ53 1 1 1 ζ5 ζ54 ζ53 ζ52 ζ54 ζ52 ζ53 ζ5 ζ52 ζ52 ζ53 ζ53 ζ5 ζ5 ζ5 ζ53 ζ54 ζ54 ζ52 ζ54 linear of order 5 ρ11 2 0 2 2 2 2 2 -1 -1 -1 0 0 0 0 2 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ12 2 0 -1 2 2 2 2 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 0 0 0 -1 -1 -1 -1 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 orthogonal lifted from D9 ρ13 2 0 -1 2 2 2 2 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 0 0 0 -1 -1 -1 -1 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 orthogonal lifted from D9 ρ14 2 0 -1 2 2 2 2 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 0 0 0 -1 -1 -1 -1 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 orthogonal lifted from D9 ρ15 2 0 2 2ζ53 2ζ5 2ζ54 2ζ52 -1 -1 -1 0 0 0 0 2ζ5 2ζ53 2ζ52 2ζ54 -ζ53 -ζ53 -ζ52 -ζ52 -ζ54 -ζ54 -ζ54 -ζ52 -ζ5 -ζ5 -ζ53 -ζ5 complex lifted from C5×S3 ρ16 2 0 2 2ζ52 2ζ54 2ζ5 2ζ53 -1 -1 -1 0 0 0 0 2ζ54 2ζ52 2ζ53 2ζ5 -ζ52 -ζ52 -ζ53 -ζ53 -ζ5 -ζ5 -ζ5 -ζ53 -ζ54 -ζ54 -ζ52 -ζ54 complex lifted from C5×S3 ρ17 2 0 2 2ζ5 2ζ52 2ζ53 2ζ54 -1 -1 -1 0 0 0 0 2ζ52 2ζ5 2ζ54 2ζ53 -ζ5 -ζ5 -ζ54 -ζ54 -ζ53 -ζ53 -ζ53 -ζ54 -ζ52 -ζ52 -ζ5 -ζ52 complex lifted from C5×S3 ρ18 2 0 2 2ζ54 2ζ53 2ζ52 2ζ5 -1 -1 -1 0 0 0 0 2ζ53 2ζ54 2ζ5 2ζ52 -ζ54 -ζ54 -ζ5 -ζ5 -ζ52 -ζ52 -ζ52 -ζ5 -ζ53 -ζ53 -ζ54 -ζ53 complex lifted from C5×S3 ρ19 2 0 -1 2ζ53 2ζ5 2ζ54 2ζ52 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 0 0 0 -ζ5 -ζ53 -ζ52 -ζ54 ζ95ζ53+ζ94ζ53 ζ98ζ53+ζ9ζ53 ζ97ζ52+ζ92ζ52 ζ95ζ52+ζ94ζ52 ζ97ζ54+ζ92ζ54 ζ95ζ54+ζ94ζ54 ζ98ζ54+ζ9ζ54 ζ98ζ52+ζ9ζ52 ζ95ζ5+ζ94ζ5 ζ98ζ5+ζ9ζ5 ζ97ζ53+ζ92ζ53 ζ97ζ5+ζ92ζ5 complex faithful ρ20 2 0 -1 2ζ53 2ζ5 2ζ54 2ζ52 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 0 0 0 -ζ5 -ζ53 -ζ52 -ζ54 ζ97ζ53+ζ92ζ53 ζ95ζ53+ζ94ζ53 ζ98ζ52+ζ9ζ52 ζ97ζ52+ζ92ζ52 ζ98ζ54+ζ9ζ54 ζ97ζ54+ζ92ζ54 ζ95ζ54+ζ94ζ54 ζ95ζ52+ζ94ζ52 ζ97ζ5+ζ92ζ5 ζ95ζ5+ζ94ζ5 ζ98ζ53+ζ9ζ53 ζ98ζ5+ζ9ζ5 complex faithful ρ21 2 0 -1 2ζ54 2ζ53 2ζ52 2ζ5 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 0 0 0 -ζ53 -ζ54 -ζ5 -ζ52 ζ97ζ54+ζ92ζ54 ζ95ζ54+ζ94ζ54 ζ98ζ5+ζ9ζ5 ζ97ζ5+ζ92ζ5 ζ98ζ52+ζ9ζ52 ζ97ζ52+ζ92ζ52 ζ95ζ52+ζ94ζ52 ζ95ζ5+ζ94ζ5 ζ97ζ53+ζ92ζ53 ζ95ζ53+ζ94ζ53 ζ98ζ54+ζ9ζ54 ζ98ζ53+ζ9ζ53 complex faithful ρ22 2 0 -1 2ζ52 2ζ54 2ζ5 2ζ53 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 0 0 0 -ζ54 -ζ52 -ζ53 -ζ5 ζ95ζ52+ζ94ζ52 ζ98ζ52+ζ9ζ52 ζ97ζ53+ζ92ζ53 ζ95ζ53+ζ94ζ53 ζ97ζ5+ζ92ζ5 ζ95ζ5+ζ94ζ5 ζ98ζ5+ζ9ζ5 ζ98ζ53+ζ9ζ53 ζ95ζ54+ζ94ζ54 ζ98ζ54+ζ9ζ54 ζ97ζ52+ζ92ζ52 ζ97ζ54+ζ92ζ54 complex faithful ρ23 2 0 -1 2ζ5 2ζ52 2ζ53 2ζ54 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 0 0 0 -ζ52 -ζ5 -ζ54 -ζ53 ζ97ζ5+ζ92ζ5 ζ95ζ5+ζ94ζ5 ζ98ζ54+ζ9ζ54 ζ97ζ54+ζ92ζ54 ζ98ζ53+ζ9ζ53 ζ97ζ53+ζ92ζ53 ζ95ζ53+ζ94ζ53 ζ95ζ54+ζ94ζ54 ζ97ζ52+ζ92ζ52 ζ95ζ52+ζ94ζ52 ζ98ζ5+ζ9ζ5 ζ98ζ52+ζ9ζ52 complex faithful ρ24 2 0 -1 2ζ5 2ζ52 2ζ53 2ζ54 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 0 0 0 -ζ52 -ζ5 -ζ54 -ζ53 ζ95ζ5+ζ94ζ5 ζ98ζ5+ζ9ζ5 ζ97ζ54+ζ92ζ54 ζ95ζ54+ζ94ζ54 ζ97ζ53+ζ92ζ53 ζ95ζ53+ζ94ζ53 ζ98ζ53+ζ9ζ53 ζ98ζ54+ζ9ζ54 ζ95ζ52+ζ94ζ52 ζ98ζ52+ζ9ζ52 ζ97ζ5+ζ92ζ5 ζ97ζ52+ζ92ζ52 complex faithful ρ25 2 0 -1 2ζ54 2ζ53 2ζ52 2ζ5 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 0 0 0 0 -ζ53 -ζ54 -ζ5 -ζ52 ζ95ζ54+ζ94ζ54 ζ98ζ54+ζ9ζ54 ζ97ζ5+ζ92ζ5 ζ95ζ5+ζ94ζ5 ζ97ζ52+ζ92ζ52 ζ95ζ52+ζ94ζ52 ζ98ζ52+ζ9ζ52 ζ98ζ5+ζ9ζ5 ζ95ζ53+ζ94ζ53 ζ98ζ53+ζ9ζ53 ζ97ζ54+ζ92ζ54 ζ97ζ53+ζ92ζ53 complex faithful ρ26 2 0 -1 2ζ53 2ζ5 2ζ54 2ζ52 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 0 0 0 -ζ5 -ζ53 -ζ52 -ζ54 ζ98ζ53+ζ9ζ53 ζ97ζ53+ζ92ζ53 ζ95ζ52+ζ94ζ52 ζ98ζ52+ζ9ζ52 ζ95ζ54+ζ94ζ54 ζ98ζ54+ζ9ζ54 ζ97ζ54+ζ92ζ54 ζ97ζ52+ζ92ζ52 ζ98ζ5+ζ9ζ5 ζ97ζ5+ζ92ζ5 ζ95ζ53+ζ94ζ53 ζ95ζ5+ζ94ζ5 complex faithful ρ27 2 0 -1 2ζ5 2ζ52 2ζ53 2ζ54 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 0 0 0 -ζ52 -ζ5 -ζ54 -ζ53 ζ98ζ5+ζ9ζ5 ζ97ζ5+ζ92ζ5 ζ95ζ54+ζ94ζ54 ζ98ζ54+ζ9ζ54 ζ95ζ53+ζ94ζ53 ζ98ζ53+ζ9ζ53 ζ97ζ53+ζ92ζ53 ζ97ζ54+ζ92ζ54 ζ98ζ52+ζ9ζ52 ζ97ζ52+ζ92ζ52 ζ95ζ5+ζ94ζ5 ζ95ζ52+ζ94ζ52 complex faithful ρ28 2 0 -1 2ζ54 2ζ53 2ζ52 2ζ5 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 0 0 0 -ζ53 -ζ54 -ζ5 -ζ52 ζ98ζ54+ζ9ζ54 ζ97ζ54+ζ92ζ54 ζ95ζ5+ζ94ζ5 ζ98ζ5+ζ9ζ5 ζ95ζ52+ζ94ζ52 ζ98ζ52+ζ9ζ52 ζ97ζ52+ζ92ζ52 ζ97ζ5+ζ92ζ5 ζ98ζ53+ζ9ζ53 ζ97ζ53+ζ92ζ53 ζ95ζ54+ζ94ζ54 ζ95ζ53+ζ94ζ53 complex faithful ρ29 2 0 -1 2ζ52 2ζ54 2ζ5 2ζ53 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 0 0 0 0 -ζ54 -ζ52 -ζ53 -ζ5 ζ97ζ52+ζ92ζ52 ζ95ζ52+ζ94ζ52 ζ98ζ53+ζ9ζ53 ζ97ζ53+ζ92ζ53 ζ98ζ5+ζ9ζ5 ζ97ζ5+ζ92ζ5 ζ95ζ5+ζ94ζ5 ζ95ζ53+ζ94ζ53 ζ97ζ54+ζ92ζ54 ζ95ζ54+ζ94ζ54 ζ98ζ52+ζ9ζ52 ζ98ζ54+ζ9ζ54 complex faithful ρ30 2 0 -1 2ζ52 2ζ54 2ζ5 2ζ53 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 0 0 0 0 -ζ54 -ζ52 -ζ53 -ζ5 ζ98ζ52+ζ9ζ52 ζ97ζ52+ζ92ζ52 ζ95ζ53+ζ94ζ53 ζ98ζ53+ζ9ζ53 ζ95ζ5+ζ94ζ5 ζ98ζ5+ζ9ζ5 ζ97ζ5+ζ92ζ5 ζ97ζ53+ζ92ζ53 ζ98ζ54+ζ9ζ54 ζ97ζ54+ζ92ζ54 ζ95ζ52+ζ94ζ52 ζ95ζ54+ζ94ζ54 complex faithful

Smallest permutation representation of C5×D9
On 45 points
Generators in S45
(1 38 29 20 11)(2 39 30 21 12)(3 40 31 22 13)(4 41 32 23 14)(5 42 33 24 15)(6 43 34 25 16)(7 44 35 26 17)(8 45 36 27 18)(9 37 28 19 10)
(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 11)(12 18)(13 17)(14 16)(19 20)(21 27)(22 26)(23 25)(28 29)(30 36)(31 35)(32 34)(37 38)(39 45)(40 44)(41 43)

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

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

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

Matrix representation of C5×D9 in GL2(𝔽181) generated by

 125 0 0 125
,
 177 131 50 127
,
 50 127 177 131
G:=sub<GL(2,GF(181))| [125,0,0,125],[177,50,131,127],[50,177,127,131] >;

C5×D9 in GAP, Magma, Sage, TeX

C_5\times D_9
% in TeX

G:=Group("C5xD9");
// GroupNames label

G:=SmallGroup(90,1);
// by ID

G=gap.SmallGroup(90,1);
# by ID

G:=PCGroup([4,-2,-5,-3,-3,602,82,963]);
// Polycyclic

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

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