direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D9×F5, D45⋊C4, D5.1D18, C5⋊(C4×D9), C9⋊F5⋊C2, C45⋊(C2×C4), (C5×D9)⋊C4, (C9×F5)⋊C2, C9⋊1(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
| C45 — D9×F5 |
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 >
5C2
9C2
45C2
5C4
45C22
45C4
3S3
5C6
15S3
9D5
9C10
45C2×C4
5C12
15Dic3
15D6
5D9
5C18
9D10
9F5
3D15
15C4×S3
5D18
5Dic9
5C36
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 |
►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
Export
Subgroup lattice of D9×F5 in TeX
Character table of D9×F5 in TeX