Copied to
clipboard

G = C5×D9order 90 = 2·32·5

Direct product of C5 and D9

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

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

Series: Derived Chief Lower central Upper central

C1C9 — C5×D9
C1C3C9C45 — C5×D9
C9 — C5×D9
C1C5

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

9C2
3S3
9C10
3C5×S3

Character table of C5×D9

 class 1235A5B5C5D9A9B9C10A10B10C10D15A15B15C15D45A45B45C45D45E45F45G45H45I45J45K45L
 size 192111122299992222222222222222
ρ1111111111111111111111111111111    trivial
ρ21-111111111-1-1-1-11111111111111111    linear of order 2
ρ31-11ζ5ζ52ζ53ζ541115352545ζ52ζ5ζ54ζ53ζ5ζ5ζ54ζ54ζ53ζ53ζ53ζ54ζ52ζ52ζ5ζ52    linear of order 10
ρ4111ζ53ζ5ζ54ζ52111ζ54ζ5ζ52ζ53ζ5ζ53ζ52ζ54ζ53ζ53ζ52ζ52ζ54ζ54ζ54ζ52ζ5ζ5ζ53ζ5    linear of order 5
ρ51-11ζ52ζ54ζ5ζ531115545352ζ54ζ52ζ53ζ5ζ52ζ52ζ53ζ53ζ5ζ5ζ5ζ53ζ54ζ54ζ52ζ54    linear of order 10
ρ6111ζ54ζ53ζ52ζ5111ζ52ζ53ζ5ζ54ζ53ζ54ζ5ζ52ζ54ζ54ζ5ζ5ζ52ζ52ζ52ζ5ζ53ζ53ζ54ζ53    linear of order 5
ρ71-11ζ53ζ5ζ54ζ521115455253ζ5ζ53ζ52ζ54ζ53ζ53ζ52ζ52ζ54ζ54ζ54ζ52ζ5ζ5ζ53ζ5    linear of order 10
ρ8111ζ5ζ52ζ53ζ54111ζ53ζ52ζ54ζ5ζ52ζ5ζ54ζ53ζ5ζ5ζ54ζ54ζ53ζ53ζ53ζ54ζ52ζ52ζ5ζ52    linear of order 5
ρ91-11ζ54ζ53ζ52ζ51115253554ζ53ζ54ζ5ζ52ζ54ζ54ζ5ζ5ζ52ζ52ζ52ζ5ζ53ζ53ζ54ζ53    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
ρ1220-12222ζ9594ζ989ζ97920000-1-1-1-1ζ989ζ9792ζ9594ζ989ζ9594ζ989ζ9792ζ9792ζ989ζ9792ζ9594ζ9594    orthogonal lifted from D9
ρ1320-12222ζ9792ζ9594ζ9890000-1-1-1-1ζ9594ζ989ζ9792ζ9594ζ9792ζ9594ζ989ζ989ζ9594ζ989ζ9792ζ9792    orthogonal lifted from D9
ρ1420-12222ζ989ζ9792ζ95940000-1-1-1-1ζ9792ζ9594ζ989ζ9792ζ989ζ9792ζ9594ζ9594ζ9792ζ9594ζ989ζ989    orthogonal lifted from D9
ρ152025355452-1-1-100005535254535352525454545255535    complex lifted from C5×S3
ρ162025254553-1-1-100005452535525253535555354545254    complex lifted from C5×S3
ρ172025525354-1-1-100005255453555454535353545252552    complex lifted from C5×S3
ρ182025453525-1-1-100005354552545455525252553535453    complex lifted from C5×S3
ρ1920-15355452ζ9792ζ9594ζ98900005535254ζ95ζ5394ζ53ζ98ζ539ζ53ζ97ζ5292ζ52ζ95ζ5294ζ52ζ97ζ5492ζ54ζ95ζ5494ζ54ζ98ζ549ζ54ζ98ζ529ζ52ζ95ζ594ζ5ζ98ζ59ζ5ζ97ζ5392ζ53ζ97ζ592ζ5    complex faithful
ρ2020-15355452ζ989ζ9792ζ959400005535254ζ97ζ5392ζ53ζ95ζ5394ζ53ζ98ζ529ζ52ζ97ζ5292ζ52ζ98ζ549ζ54ζ97ζ5492ζ54ζ95ζ5494ζ54ζ95ζ5294ζ52ζ97ζ592ζ5ζ95ζ594ζ5ζ98ζ539ζ53ζ98ζ59ζ5    complex faithful
ρ2120-15453525ζ989ζ9792ζ959400005354552ζ97ζ5492ζ54ζ95ζ5494ζ54ζ98ζ59ζ5ζ97ζ592ζ5ζ98ζ529ζ52ζ97ζ5292ζ52ζ95ζ5294ζ52ζ95ζ594ζ5ζ97ζ5392ζ53ζ95ζ5394ζ53ζ98ζ549ζ54ζ98ζ539ζ53    complex faithful
ρ2220-15254553ζ9792ζ9594ζ98900005452535ζ95ζ5294ζ52ζ98ζ529ζ52ζ97ζ5392ζ53ζ95ζ5394ζ53ζ97ζ592ζ5ζ95ζ594ζ5ζ98ζ59ζ5ζ98ζ539ζ53ζ95ζ5494ζ54ζ98ζ549ζ54ζ97ζ5292ζ52ζ97ζ5492ζ54    complex faithful
ρ2320-15525354ζ989ζ9792ζ959400005255453ζ97ζ592ζ5ζ95ζ594ζ5ζ98ζ549ζ54ζ97ζ5492ζ54ζ98ζ539ζ53ζ97ζ5392ζ53ζ95ζ5394ζ53ζ95ζ5494ζ54ζ97ζ5292ζ52ζ95ζ5294ζ52ζ98ζ59ζ5ζ98ζ529ζ52    complex faithful
ρ2420-15525354ζ9792ζ9594ζ98900005255453ζ95ζ594ζ5ζ98ζ59ζ5ζ97ζ5492ζ54ζ95ζ5494ζ54ζ97ζ5392ζ53ζ95ζ5394ζ53ζ98ζ539ζ53ζ98ζ549ζ54ζ95ζ5294ζ52ζ98ζ529ζ52ζ97ζ592ζ5ζ97ζ5292ζ52    complex faithful
ρ2520-15453525ζ9792ζ9594ζ98900005354552ζ95ζ5494ζ54ζ98ζ549ζ54ζ97ζ592ζ5ζ95ζ594ζ5ζ97ζ5292ζ52ζ95ζ5294ζ52ζ98ζ529ζ52ζ98ζ59ζ5ζ95ζ5394ζ53ζ98ζ539ζ53ζ97ζ5492ζ54ζ97ζ5392ζ53    complex faithful
ρ2620-15355452ζ9594ζ989ζ979200005535254ζ98ζ539ζ53ζ97ζ5392ζ53ζ95ζ5294ζ52ζ98ζ529ζ52ζ95ζ5494ζ54ζ98ζ549ζ54ζ97ζ5492ζ54ζ97ζ5292ζ52ζ98ζ59ζ5ζ97ζ592ζ5ζ95ζ5394ζ53ζ95ζ594ζ5    complex faithful
ρ2720-15525354ζ9594ζ989ζ979200005255453ζ98ζ59ζ5ζ97ζ592ζ5ζ95ζ5494ζ54ζ98ζ549ζ54ζ95ζ5394ζ53ζ98ζ539ζ53ζ97ζ5392ζ53ζ97ζ5492ζ54ζ98ζ529ζ52ζ97ζ5292ζ52ζ95ζ594ζ5ζ95ζ5294ζ52    complex faithful
ρ2820-15453525ζ9594ζ989ζ979200005354552ζ98ζ549ζ54ζ97ζ5492ζ54ζ95ζ594ζ5ζ98ζ59ζ5ζ95ζ5294ζ52ζ98ζ529ζ52ζ97ζ5292ζ52ζ97ζ592ζ5ζ98ζ539ζ53ζ97ζ5392ζ53ζ95ζ5494ζ54ζ95ζ5394ζ53    complex faithful
ρ2920-15254553ζ989ζ9792ζ959400005452535ζ97ζ5292ζ52ζ95ζ5294ζ52ζ98ζ539ζ53ζ97ζ5392ζ53ζ98ζ59ζ5ζ97ζ592ζ5ζ95ζ594ζ5ζ95ζ5394ζ53ζ97ζ5492ζ54ζ95ζ5494ζ54ζ98ζ529ζ52ζ98ζ549ζ54    complex faithful
ρ3020-15254553ζ9594ζ989ζ979200005452535ζ98ζ529ζ52ζ97ζ5292ζ52ζ95ζ5394ζ53ζ98ζ539ζ53ζ95ζ594ζ5ζ98ζ59ζ5ζ97ζ592ζ5ζ97ζ5392ζ53ζ98ζ549ζ54ζ97ζ5492ζ54ζ95ζ5294ζ52ζ95ζ5494ζ54    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

׿
×
𝔽