metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C9⋊F5, C5⋊Dic9, C45⋊1C4, D5.D9, C15.Dic3, C3.(C3⋊F5), (C9×D5).1C2, (C3×D5).2S3, SmallGroup(180,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C45 — C9⋊F5 |
Generators and relations for C9⋊F5
G = < a,b,c | a9=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >
Character table of C9⋊F5
| class | 1 | 2 | 3 | 4A | 4B | 5 | 6 | 9A | 9B | 9C | 15A | 15B | 18A | 18B | 18C | 45A | 45B | 45C | 45D | 45E | 45F | |
| size | 1 | 5 | 2 | 45 | 45 | 4 | 10 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | -i | i | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | 1 | i | -i | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | -1 | 0 | 0 | 2 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ7 | 2 | 2 | -1 | 0 | 0 | 2 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ8 | 2 | 2 | -1 | 0 | 0 | 2 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | -2 | -1 | 0 | 0 | 2 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | symplectic lifted from Dic9, Schur index 2 |
| ρ10 | 2 | -2 | -1 | 0 | 0 | 2 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | symplectic lifted from Dic9, Schur index 2 |
| ρ11 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | -1 | -1 | -1 | 2 | 2 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ12 | 2 | -2 | -1 | 0 | 0 | 2 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | symplectic lifted from Dic9, Schur index 2 |
| ρ13 | 4 | 0 | 4 | 0 | 0 | -1 | 0 | 4 | 4 | 4 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 0 | 4 | 0 | 0 | -1 | 0 | -2 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | complex lifted from C3⋊F5 |
| ρ15 | 4 | 0 | 4 | 0 | 0 | -1 | 0 | -2 | -2 | -2 | -1 | -1 | 0 | 0 | 0 | 1+√-15/2 | 1+√-15/2 | 1-√-15/2 | 1-√-15/2 | 1-√-15/2 | 1+√-15/2 | complex lifted from C3⋊F5 |
| ρ16 | 4 | 0 | -2 | 0 | 0 | -1 | 0 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 1+√-15/2 | 1-√-15/2 | 0 | 0 | 0 | -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 | -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 | -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 | -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 | ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 | ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 | complex faithful |
| ρ17 | 4 | 0 | -2 | 0 | 0 | -1 | 0 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 1-√-15/2 | 1+√-15/2 | 0 | 0 | 0 | -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 | ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 | ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 | -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 | -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 | -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 | complex faithful |
| ρ18 | 4 | 0 | -2 | 0 | 0 | -1 | 0 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 1+√-15/2 | 1-√-15/2 | 0 | 0 | 0 | ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 | -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 | ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 | -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 | -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 | -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 | complex faithful |
| ρ19 | 4 | 0 | -2 | 0 | 0 | -1 | 0 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 1+√-15/2 | 1-√-15/2 | 0 | 0 | 0 | -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 | ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 | -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 | ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 | -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 | -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 | complex faithful |
| ρ20 | 4 | 0 | -2 | 0 | 0 | -1 | 0 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 2ζ98+2ζ9 | 1-√-15/2 | 1+√-15/2 | 0 | 0 | 0 | -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 | -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 | -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 | ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 | -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 | ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 | complex faithful |
| ρ21 | 4 | 0 | -2 | 0 | 0 | -1 | 0 | 2ζ98+2ζ9 | 2ζ97+2ζ92 | 2ζ95+2ζ94 | 1-√-15/2 | 1+√-15/2 | 0 | 0 | 0 | ζ95ζ53+ζ95ζ52-ζ94ζ53-ζ94ζ52-ζ94 | -ζ98ζ54-ζ98ζ5-ζ98+ζ9ζ54+ζ9ζ5 | -ζ97ζ54-ζ97ζ5-ζ97+ζ92ζ54+ζ92ζ5 | -ζ95ζ53-ζ95ζ52-ζ95+ζ94ζ53+ζ94ζ52 | ζ98ζ54+ζ98ζ5-ζ9ζ54-ζ9ζ5-ζ9 | -ζ97ζ53-ζ97ζ52-ζ97+ζ92ζ53+ζ92ζ52 | complex 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 10 24 43 33)(2 11 25 44 34)(3 12 26 45 35)(4 13 27 37 36)(5 14 19 38 28)(6 15 20 39 29)(7 16 21 40 30)(8 17 22 41 31)(9 18 23 42 32)
(2 9)(3 8)(4 7)(5 6)(10 24 33 43)(11 23 34 42)(12 22 35 41)(13 21 36 40)(14 20 28 39)(15 19 29 38)(16 27 30 37)(17 26 31 45)(18 25 32 44)
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), (1,10,24,43,33)(2,11,25,44,34)(3,12,26,45,35)(4,13,27,37,36)(5,14,19,38,28)(6,15,20,39,29)(7,16,21,40,30)(8,17,22,41,31)(9,18,23,42,32), (2,9)(3,8)(4,7)(5,6)(10,24,33,43)(11,23,34,42)(12,22,35,41)(13,21,36,40)(14,20,28,39)(15,19,29,38)(16,27,30,37)(17,26,31,45)(18,25,32,44)>;
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), (1,10,24,43,33)(2,11,25,44,34)(3,12,26,45,35)(4,13,27,37,36)(5,14,19,38,28)(6,15,20,39,29)(7,16,21,40,30)(8,17,22,41,31)(9,18,23,42,32), (2,9)(3,8)(4,7)(5,6)(10,24,33,43)(11,23,34,42)(12,22,35,41)(13,21,36,40)(14,20,28,39)(15,19,29,38)(16,27,30,37)(17,26,31,45)(18,25,32,44) );
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)], [(1,10,24,43,33),(2,11,25,44,34),(3,12,26,45,35),(4,13,27,37,36),(5,14,19,38,28),(6,15,20,39,29),(7,16,21,40,30),(8,17,22,41,31),(9,18,23,42,32)], [(2,9),(3,8),(4,7),(5,6),(10,24,33,43),(11,23,34,42),(12,22,35,41),(13,21,36,40),(14,20,28,39),(15,19,29,38),(16,27,30,37),(17,26,31,45),(18,25,32,44)]])
C9⋊F5 is a maximal subgroup of
D9×F5
C9⋊F5 is a maximal quotient of C45⋊C8
Matrix representation of C9⋊F5 ►in GL4(𝔽181) generated by
| 4 | 54 | 0 | 0 |
| 127 | 131 | 0 | 0 |
| 0 | 0 | 4 | 54 |
| 0 | 0 | 127 | 131 |
| 74 | 149 | 180 | 0 |
| 32 | 106 | 0 | 180 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 180 | 180 | 0 | 0 |
| 107 | 32 | 107 | 32 |
| 106 | 74 | 106 | 74 |
G:=sub<GL(4,GF(181))| [4,127,0,0,54,131,0,0,0,0,4,127,0,0,54,131],[74,32,1,0,149,106,0,1,180,0,0,0,0,180,0,0],[1,180,107,106,0,180,32,74,0,0,107,106,0,0,32,74] >;
C9⋊F5 in GAP, Magma, Sage, TeX
C_9\rtimes F_5
% in TeX
G:=Group("C9:F5"); // GroupNames label
G:=SmallGroup(180,6);
// by ID
G=gap.SmallGroup(180,6);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-3,10,1022,462,723,488,3004]);
// Polycyclic
G:=Group<a,b,c|a^9=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C9⋊F5 in TeX
Character table of C9⋊F5 in TeX