Copied to
clipboard

G = D45order 90 = 2·32·5

Dihedral group

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

Aliases: D45, C9⋊D5, C5⋊D9, C451C2, C3.D15, C15.1S3, sometimes denoted D90 or Dih45 or Dih90, SmallGroup(90,3)

Series: Derived Chief Lower central Upper central

C1C45 — D45
C1C3C15C45 — D45
C45 — D45
C1

Generators and relations for D45
 G = < a,b | a45=b2=1, bab=a-1 >

45C2
15S3
9D5
5D9
3D15

Character table of D45

 class 1235A5B9A9B9C15A15B15C15D45A45B45C45D45E45F45G45H45I45J45K45L
 size 1452222222222222222222222
ρ1111111111111111111111111    trivial
ρ21-11111111111111111111111    linear of order 2
ρ320222-1-1-12222-1-1-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ4202-1+5/2-1-5/2222-1+5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2-1+5/2    orthogonal lifted from D5
ρ5202-1-5/2-1+5/2222-1-5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1+5/2-1+5/2-1+5/2-1+5/2-1+5/2-1+5/2-1-5/2-1-5/2-1-5/2-1-5/2    orthogonal lifted from D5
ρ620-122ζ9792ζ9594ζ989-1-1-1-1ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ720-122ζ989ζ9792ζ9594-1-1-1-1ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ820-122ζ9594ζ989ζ9792-1-1-1-1ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ9202-1-5/2-1+5/2-1-1-1-1-5/2-1+5/2-1+5/2-1-5/23ζ533ζ52533ζ533ζ5253ζ3ζ543ζ55ζ3ζ543ζ55ζ3ζ543ζ55ζ32ζ5432ζ55ζ32ζ5432ζ55ζ32ζ5432ζ55ζ3ζ533ζ5252ζ3ζ533ζ5252ζ3ζ533ζ52523ζ533ζ5253    orthogonal lifted from D15
ρ10202-1+5/2-1-5/2-1-1-1-1+5/2-1-5/2-1-5/2-1+5/2ζ3ζ543ζ55ζ3ζ543ζ55ζ3ζ533ζ5252ζ3ζ533ζ5252ζ3ζ533ζ52523ζ533ζ52533ζ533ζ52533ζ533ζ5253ζ32ζ5432ζ55ζ32ζ5432ζ55ζ32ζ5432ζ55ζ3ζ543ζ55    orthogonal lifted from D15
ρ11202-1+5/2-1-5/2-1-1-1-1+5/2-1-5/2-1-5/2-1+5/2ζ32ζ5432ζ55ζ32ζ5432ζ553ζ533ζ52533ζ533ζ52533ζ533ζ5253ζ3ζ533ζ5252ζ3ζ533ζ5252ζ3ζ533ζ5252ζ3ζ543ζ55ζ3ζ543ζ55ζ3ζ543ζ55ζ32ζ5432ζ55    orthogonal lifted from D15
ρ12202-1-5/2-1+5/2-1-1-1-1-5/2-1+5/2-1+5/2-1-5/2ζ3ζ533ζ5252ζ3ζ533ζ5252ζ32ζ5432ζ55ζ32ζ5432ζ55ζ32ζ5432ζ55ζ3ζ543ζ55ζ3ζ543ζ55ζ3ζ543ζ553ζ533ζ52533ζ533ζ52533ζ533ζ5253ζ3ζ533ζ5252    orthogonal lifted from D15
ρ1320-1-1-5/2-1+5/2ζ9792ζ9594ζ989ζ93ζ5393ζ525296ζ5496ζ55493ζ5493ζ55493ζ5393ζ5253ζ98ζ539ζ52ζ97ζ5292ζ53ζ95ζ594ζ54ζ98ζ59ζ54ζ97ζ5492ζ5ζ95ζ5494ζ5ζ98ζ549ζ5ζ97ζ592ζ54ζ95ζ5294ζ53ζ98ζ529ζ53ζ97ζ5392ζ52ζ95ζ5394ζ52    orthogonal faithful
ρ1420-1-1+5/2-1-5/2ζ989ζ9792ζ959493ζ5493ζ554ζ93ζ5393ζ525293ζ5393ζ525396ζ5496ζ554ζ95ζ594ζ54ζ98ζ59ζ54ζ97ζ5392ζ52ζ95ζ5294ζ53ζ98ζ529ζ53ζ97ζ5292ζ53ζ95ζ5394ζ52ζ98ζ539ζ52ζ97ζ592ζ54ζ95ζ5494ζ5ζ98ζ549ζ5ζ97ζ5492ζ5    orthogonal faithful
ρ1520-1-1+5/2-1-5/2ζ9594ζ989ζ979296ζ5496ζ55493ζ5393ζ5253ζ93ζ5393ζ525293ζ5493ζ554ζ97ζ592ζ54ζ95ζ5494ζ5ζ98ζ539ζ52ζ97ζ5292ζ53ζ95ζ5394ζ52ζ98ζ529ζ53ζ97ζ5392ζ52ζ95ζ5294ζ53ζ98ζ59ζ54ζ97ζ5492ζ5ζ95ζ594ζ54ζ98ζ549ζ5    orthogonal faithful
ρ1620-1-1-5/2-1+5/2ζ9594ζ989ζ9792ζ93ζ5393ζ525296ζ5496ζ55493ζ5493ζ55493ζ5393ζ5253ζ97ζ5292ζ53ζ95ζ5394ζ52ζ98ζ59ζ54ζ97ζ5492ζ5ζ95ζ594ζ54ζ98ζ549ζ5ζ97ζ592ζ54ζ95ζ5494ζ5ζ98ζ529ζ53ζ97ζ5392ζ52ζ95ζ5294ζ53ζ98ζ539ζ52    orthogonal faithful
ρ1720-1-1-5/2-1+5/2ζ989ζ9792ζ959493ζ5393ζ525393ζ5493ζ55496ζ5496ζ554ζ93ζ5393ζ5252ζ95ζ5294ζ53ζ98ζ529ζ53ζ97ζ592ζ54ζ95ζ5494ζ5ζ98ζ549ζ5ζ97ζ5492ζ5ζ95ζ594ζ54ζ98ζ59ζ54ζ97ζ5292ζ53ζ95ζ5394ζ52ζ98ζ539ζ52ζ97ζ5392ζ52    orthogonal faithful
ρ1820-1-1+5/2-1-5/2ζ9792ζ9594ζ98996ζ5496ζ55493ζ5393ζ5253ζ93ζ5393ζ525293ζ5493ζ554ζ98ζ549ζ5ζ97ζ592ζ54ζ95ζ5394ζ52ζ98ζ539ζ52ζ97ζ5292ζ53ζ95ζ5294ζ53ζ98ζ529ζ53ζ97ζ5392ζ52ζ95ζ594ζ54ζ98ζ59ζ54ζ97ζ5492ζ5ζ95ζ5494ζ5    orthogonal faithful
ρ1920-1-1+5/2-1-5/2ζ9792ζ9594ζ98993ζ5493ζ554ζ93ζ5393ζ525293ζ5393ζ525396ζ5496ζ554ζ98ζ59ζ54ζ97ζ5492ζ5ζ95ζ5294ζ53ζ98ζ529ζ53ζ97ζ5392ζ52ζ95ζ5394ζ52ζ98ζ539ζ52ζ97ζ5292ζ53ζ95ζ5494ζ5ζ98ζ549ζ5ζ97ζ592ζ54ζ95ζ594ζ54    orthogonal faithful
ρ2020-1-1-5/2-1+5/2ζ9594ζ989ζ979293ζ5393ζ525393ζ5493ζ55496ζ5496ζ554ζ93ζ5393ζ5252ζ97ζ5392ζ52ζ95ζ5294ζ53ζ98ζ549ζ5ζ97ζ592ζ54ζ95ζ5494ζ5ζ98ζ59ζ54ζ97ζ5492ζ5ζ95ζ594ζ54ζ98ζ539ζ52ζ97ζ5292ζ53ζ95ζ5394ζ52ζ98ζ529ζ53    orthogonal faithful
ρ2120-1-1-5/2-1+5/2ζ9792ζ9594ζ98993ζ5393ζ525393ζ5493ζ55496ζ5496ζ554ζ93ζ5393ζ5252ζ98ζ529ζ53ζ97ζ5392ζ52ζ95ζ5494ζ5ζ98ζ549ζ5ζ97ζ592ζ54ζ95ζ594ζ54ζ98ζ59ζ54ζ97ζ5492ζ5ζ95ζ5394ζ52ζ98ζ539ζ52ζ97ζ5292ζ53ζ95ζ5294ζ53    orthogonal faithful
ρ2220-1-1+5/2-1-5/2ζ9594ζ989ζ979293ζ5493ζ554ζ93ζ5393ζ525293ζ5393ζ525396ζ5496ζ554ζ97ζ5492ζ5ζ95ζ594ζ54ζ98ζ529ζ53ζ97ζ5392ζ52ζ95ζ5294ζ53ζ98ζ539ζ52ζ97ζ5292ζ53ζ95ζ5394ζ52ζ98ζ549ζ5ζ97ζ592ζ54ζ95ζ5494ζ5ζ98ζ59ζ54    orthogonal faithful
ρ2320-1-1-5/2-1+5/2ζ989ζ9792ζ9594ζ93ζ5393ζ525296ζ5496ζ55493ζ5493ζ55493ζ5393ζ5253ζ95ζ5394ζ52ζ98ζ539ζ52ζ97ζ5492ζ5ζ95ζ594ζ54ζ98ζ59ζ54ζ97ζ592ζ54ζ95ζ5494ζ5ζ98ζ549ζ5ζ97ζ5392ζ52ζ95ζ5294ζ53ζ98ζ529ζ53ζ97ζ5292ζ53    orthogonal faithful
ρ2420-1-1+5/2-1-5/2ζ989ζ9792ζ959496ζ5496ζ55493ζ5393ζ5253ζ93ζ5393ζ525293ζ5493ζ554ζ95ζ5494ζ5ζ98ζ549ζ5ζ97ζ5292ζ53ζ95ζ5394ζ52ζ98ζ539ζ52ζ97ζ5392ζ52ζ95ζ5294ζ53ζ98ζ529ζ53ζ97ζ5492ζ5ζ95ζ594ζ54ζ98ζ59ζ54ζ97ζ592ζ54    orthogonal faithful

Smallest permutation representation of D45
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)
(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 25)(23 24)

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

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

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

D45 is a maximal subgroup of   D5×D9  D135  D45⋊C3  C3⋊D45  C22⋊D45  D225  C5⋊D45
D45 is a maximal quotient of   Dic45  D135  C3⋊D45  C22⋊D45  D225  C5⋊D45

Matrix representation of D45 in GL4(𝔽181) generated by

1775000
13112700
0032167
0045167
,
1000
18018000
0010
0055180
G:=sub<GL(4,GF(181))| [177,131,0,0,50,127,0,0,0,0,32,45,0,0,167,167],[1,180,0,0,0,180,0,0,0,0,1,55,0,0,0,180] >;

D45 in GAP, Magma, Sage, TeX

D_{45}
% in TeX

G:=Group("D45");
// GroupNames label

G:=SmallGroup(90,3);
// by ID

G=gap.SmallGroup(90,3);
# by ID

G:=PCGroup([4,-2,-3,-5,-3,273,245,290,963]);
// Polycyclic

G:=Group<a,b|a^45=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D45 in TeX
Character table of D45 in TeX

׿
×
𝔽