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G = C5×D9  order 90 = 2·32·5

Direct product of C5 and D9

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C5×D9, C9⋊C10, C45⋊2C2, 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 >

C1
9C2
C3
C5
3S3
C9
9C10
C15
D9
3C5×S3
C45
C5×D9

Character table of C5×D9

 class 1235A5B5C5D9A9B9C10A10B10C10D15A15B15C15D45A45B45C45D45E45F45G45H45I45J45K45L
 size 192111122299992222222222222222
ρ1111111111111111111111111111111    trivial
ρ21-111111111-1-1-1-11111111111111111    linear of order 2
ρ31-11ζ5ζ52ζ53ζ54111-ζ53-ζ52-ζ54-ζ5ζ52ζ5ζ54ζ53ζ5ζ5ζ54ζ54ζ53ζ53ζ53ζ54ζ52ζ52ζ5ζ52    linear of order 10
ρ4111ζ53ζ5ζ54ζ52111ζ54ζ5ζ52ζ53ζ5ζ53ζ52ζ54ζ53ζ53ζ52ζ52ζ54ζ54ζ54ζ52ζ5ζ5ζ53ζ5    linear of order 5
ρ51-11ζ52ζ54ζ5ζ53111-ζ5-ζ54-ζ53-ζ52ζ54ζ52ζ53ζ5ζ52ζ52ζ53ζ53ζ5ζ5ζ5ζ53ζ54ζ54ζ52ζ54    linear of order 10
ρ6111ζ54ζ53ζ52ζ5111ζ52ζ53ζ5ζ54ζ53ζ54ζ5ζ52ζ54ζ54ζ5ζ5ζ52ζ52ζ52ζ5ζ53ζ53ζ54ζ53    linear of order 5
ρ71-11ζ53ζ5ζ54ζ52111-ζ54-ζ5-ζ52-ζ53ζ5ζ53ζ52ζ54ζ53ζ53ζ52ζ52ζ54ζ54ζ54ζ52ζ5ζ5ζ53ζ5    linear of order 10
ρ8111ζ5ζ52ζ53ζ54111ζ53ζ52ζ54ζ5ζ52ζ5ζ54ζ53ζ5ζ5ζ54ζ54ζ53ζ53ζ53ζ54ζ52ζ52ζ5ζ52    linear of order 5
ρ91-11ζ54ζ53ζ52ζ5111-ζ52-ζ53-ζ5-ζ54ζ53ζ54ζ5ζ52ζ54ζ54ζ5ζ5ζ52ζ52ζ52ζ5ζ53ζ53ζ54ζ53    linear of order 10
ρ10111ζ52ζ54ζ5ζ53111ζ5ζ54ζ53ζ52ζ54ζ52ζ53ζ5ζ52ζ52ζ53ζ53ζ5ζ5ζ5ζ53ζ54ζ54ζ52ζ54    linear of order 5
ρ112022222-1-1-100002222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ1220-12222ζ95+ζ94ζ98+ζ9ζ97+ζ920000-1-1-1-1ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94    orthogonal lifted from D9
ρ1320-12222ζ97+ζ92ζ95+ζ94ζ98+ζ90000-1-1-1-1ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ97+ζ92    orthogonal lifted from D9
ρ1420-12222ζ98+ζ9ζ97+ζ92ζ95+ζ940000-1-1-1-1ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ95+ζ94ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ98+ζ9    orthogonal lifted from D9
ρ152022ζ532ζ52ζ542ζ52-1-1-100002ζ52ζ532ζ522ζ54-ζ53-ζ53-ζ52-ζ52-ζ54-ζ54-ζ54-ζ52-ζ5-ζ5-ζ53-ζ5    complex lifted from C5×S3
ρ162022ζ522ζ542ζ52ζ53-1-1-100002ζ542ζ522ζ532ζ5-ζ52-ζ52-ζ53-ζ53-ζ5-ζ5-ζ5-ζ53-ζ54-ζ54-ζ52-ζ54    complex lifted from C5×S3
ρ172022ζ52ζ522ζ532ζ54-1-1-100002ζ522ζ52ζ542ζ53-ζ5-ζ5-ζ54-ζ54-ζ53-ζ53-ζ53-ζ54-ζ52-ζ52-ζ5-ζ52    complex lifted from C5×S3
ρ182022ζ542ζ532ζ522ζ5-1-1-100002ζ532ζ542ζ52ζ52-ζ54-ζ54-ζ5-ζ5-ζ52-ζ52-ζ52-ζ5-ζ53-ζ53-ζ54-ζ53    complex lifted from C5×S3
ρ1920-12ζ532ζ52ζ542ζ52ζ97+ζ92ζ95+ζ94ζ98+ζ90000-ζ5-ζ53-ζ52-ζ54ζ95ζ53+ζ94ζ53ζ98ζ53+ζ9ζ53ζ97ζ52+ζ92ζ52ζ95ζ52+ζ94ζ52ζ97ζ54+ζ92ζ54ζ95ζ54+ζ94ζ54ζ98ζ54+ζ9ζ54ζ98ζ52+ζ9ζ52ζ95ζ5+ζ94ζ5ζ98ζ5+ζ9ζ5ζ97ζ53+ζ92ζ53ζ97ζ5+ζ92ζ5    complex faithful
ρ2020-12ζ532ζ52ζ542ζ52ζ98+ζ9ζ97+ζ92ζ95+ζ940000-ζ5-ζ53-ζ52-ζ54ζ97ζ53+ζ92ζ53ζ95ζ53+ζ94ζ53ζ98ζ52+ζ9ζ52ζ97ζ52+ζ92ζ52ζ98ζ54+ζ9ζ54ζ97ζ54+ζ92ζ54ζ95ζ54+ζ94ζ54ζ95ζ52+ζ94ζ52ζ97ζ5+ζ92ζ5ζ95ζ5+ζ94ζ5ζ98ζ53+ζ9ζ53ζ98ζ5+ζ9ζ5    complex faithful
ρ2120-12ζ542ζ532ζ522ζ5ζ98+ζ9ζ97+ζ92ζ95+ζ940000-ζ53-ζ54-ζ5-ζ52ζ97ζ54+ζ92ζ54ζ95ζ54+ζ94ζ54ζ98ζ5+ζ9ζ5ζ97ζ5+ζ92ζ5ζ98ζ52+ζ9ζ52ζ97ζ52+ζ92ζ52ζ95ζ52+ζ94ζ52ζ95ζ5+ζ94ζ5ζ97ζ53+ζ92ζ53ζ95ζ53+ζ94ζ53ζ98ζ54+ζ9ζ54ζ98ζ53+ζ9ζ53    complex faithful
ρ2220-12ζ522ζ542ζ52ζ53ζ97+ζ92ζ95+ζ94ζ98+ζ90000-ζ54-ζ52-ζ53-ζ5ζ95ζ52+ζ94ζ52ζ98ζ52+ζ9ζ52ζ97ζ53+ζ92ζ53ζ95ζ53+ζ94ζ53ζ97ζ5+ζ92ζ5ζ95ζ5+ζ94ζ5ζ98ζ5+ζ9ζ5ζ98ζ53+ζ9ζ53ζ95ζ54+ζ94ζ54ζ98ζ54+ζ9ζ54ζ97ζ52+ζ92ζ52ζ97ζ54+ζ92ζ54    complex faithful
ρ2320-12ζ52ζ522ζ532ζ54ζ98+ζ9ζ97+ζ92ζ95+ζ940000-ζ52-ζ5-ζ54-ζ53ζ97ζ5+ζ92ζ5ζ95ζ5+ζ94ζ5ζ98ζ54+ζ9ζ54ζ97ζ54+ζ92ζ54ζ98ζ53+ζ9ζ53ζ97ζ53+ζ92ζ53ζ95ζ53+ζ94ζ53ζ95ζ54+ζ94ζ54ζ97ζ52+ζ92ζ52ζ95ζ52+ζ94ζ52ζ98ζ5+ζ9ζ5ζ98ζ52+ζ9ζ52    complex faithful
ρ2420-12ζ52ζ522ζ532ζ54ζ97+ζ92ζ95+ζ94ζ98+ζ90000-ζ52-ζ5-ζ54-ζ53ζ95ζ5+ζ94ζ5ζ98ζ5+ζ9ζ5ζ97ζ54+ζ92ζ54ζ95ζ54+ζ94ζ54ζ97ζ53+ζ92ζ53ζ95ζ53+ζ94ζ53ζ98ζ53+ζ9ζ53ζ98ζ54+ζ9ζ54ζ95ζ52+ζ94ζ52ζ98ζ52+ζ9ζ52ζ97ζ5+ζ92ζ5ζ97ζ52+ζ92ζ52    complex faithful
ρ2520-12ζ542ζ532ζ522ζ5ζ97+ζ92ζ95+ζ94ζ98+ζ90000-ζ53-ζ54-ζ5-ζ52ζ95ζ54+ζ94ζ54ζ98ζ54+ζ9ζ54ζ97ζ5+ζ92ζ5ζ95ζ5+ζ94ζ5ζ97ζ52+ζ92ζ52ζ95ζ52+ζ94ζ52ζ98ζ52+ζ9ζ52ζ98ζ5+ζ9ζ5ζ95ζ53+ζ94ζ53ζ98ζ53+ζ9ζ53ζ97ζ54+ζ92ζ54ζ97ζ53+ζ92ζ53    complex faithful
ρ2620-12ζ532ζ52ζ542ζ52ζ95+ζ94ζ98+ζ9ζ97+ζ920000-ζ5-ζ53-ζ52-ζ54ζ98ζ53+ζ9ζ53ζ97ζ53+ζ92ζ53ζ95ζ52+ζ94ζ52ζ98ζ52+ζ9ζ52ζ95ζ54+ζ94ζ54ζ98ζ54+ζ9ζ54ζ97ζ54+ζ92ζ54ζ97ζ52+ζ92ζ52ζ98ζ5+ζ9ζ5ζ97ζ5+ζ92ζ5ζ95ζ53+ζ94ζ53ζ95ζ5+ζ94ζ5    complex faithful
ρ2720-12ζ52ζ522ζ532ζ54ζ95+ζ94ζ98+ζ9ζ97+ζ920000-ζ52-ζ5-ζ54-ζ53ζ98ζ5+ζ9ζ5ζ97ζ5+ζ92ζ5ζ95ζ54+ζ94ζ54ζ98ζ54+ζ9ζ54ζ95ζ53+ζ94ζ53ζ98ζ53+ζ9ζ53ζ97ζ53+ζ92ζ53ζ97ζ54+ζ92ζ54ζ98ζ52+ζ9ζ52ζ97ζ52+ζ92ζ52ζ95ζ5+ζ94ζ5ζ95ζ52+ζ94ζ52    complex faithful
ρ2820-12ζ542ζ532ζ522ζ5ζ95+ζ94ζ98+ζ9ζ97+ζ920000-ζ53-ζ54-ζ5-ζ52ζ98ζ54+ζ9ζ54ζ97ζ54+ζ92ζ54ζ95ζ5+ζ94ζ5ζ98ζ5+ζ9ζ5ζ95ζ52+ζ94ζ52ζ98ζ52+ζ9ζ52ζ97ζ52+ζ92ζ52ζ97ζ5+ζ92ζ5ζ98ζ53+ζ9ζ53ζ97ζ53+ζ92ζ53ζ95ζ54+ζ94ζ54ζ95ζ53+ζ94ζ53    complex faithful
ρ2920-12ζ522ζ542ζ52ζ53ζ98+ζ9ζ97+ζ92ζ95+ζ940000-ζ54-ζ52-ζ53-ζ5ζ97ζ52+ζ92ζ52ζ95ζ52+ζ94ζ52ζ98ζ53+ζ9ζ53ζ97ζ53+ζ92ζ53ζ98ζ5+ζ9ζ5ζ97ζ5+ζ92ζ5ζ95ζ5+ζ94ζ5ζ95ζ53+ζ94ζ53ζ97ζ54+ζ92ζ54ζ95ζ54+ζ94ζ54ζ98ζ52+ζ9ζ52ζ98ζ54+ζ9ζ54    complex faithful
ρ3020-12ζ522ζ542ζ52ζ53ζ95+ζ94ζ98+ζ9ζ97+ζ920000-ζ54-ζ52-ζ53-ζ5ζ98ζ52+ζ9ζ52ζ97ζ52+ζ92ζ52ζ95ζ53+ζ94ζ53ζ98ζ53+ζ9ζ53ζ95ζ5+ζ94ζ5ζ98ζ5+ζ9ζ5ζ97ζ5+ζ92ζ5ζ97ζ53+ζ92ζ53ζ98ζ54+ζ9ζ54ζ97ζ54+ζ92ζ54ζ95ζ52+ζ94ζ52ζ95ζ54+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

1250
0125
,
177131
50127
,
50127
177131
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
 

Export

Subgroup lattice of C5×D9 in TeX
Character table of C5×D9 in TeX

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