direct product, metabelian, supersoluble, monomial, A-group
Aliases: D5×F5, D5.1D10, C5⋊D5⋊C4, C5⋊(C4×D5), (C5×F5)⋊C2, D52.1C2, D5.D5⋊C2, C5⋊4(C2×F5), (C5×D5)⋊1C4, C52⋊1(C2×C4), (C5×D5).C22, SmallGroup(200,41)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — D5×F5 |
Generators and relations for D5×F5
G = < a,b,c,d | a5=b2=c5=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Character table of D5×F5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | 10A | 10B | 10C | 20A | 20B | 20C | 20D | |
| size | 1 | 5 | 5 | 25 | 5 | 5 | 25 | 25 | 2 | 2 | 4 | 8 | 8 | 10 | 10 | 20 | 10 | 10 | 10 | 10 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 0 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | orthogonal lifted from D5 |
| ρ10 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 0 | 1+√5/2 | 1-√5/2 | 1+√5/2 | 1-√5/2 | orthogonal lifted from D10 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | 0 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | orthogonal lifted from D5 |
| ρ12 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | 0 | 1-√5/2 | 1+√5/2 | 1-√5/2 | 1+√5/2 | orthogonal lifted from D10 |
| ρ13 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 0 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | complex lifted from C4×D5 |
| ρ14 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 0 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | complex lifted from C4×D5 |
| ρ15 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | 0 | ζ43ζ53+ζ43ζ52 | ζ43ζ54+ζ43ζ5 | ζ4ζ53+ζ4ζ52 | ζ4ζ54+ζ4ζ5 | complex lifted from C4×D5 |
| ρ16 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | 0 | ζ43ζ54+ζ43ζ5 | ζ43ζ53+ζ43ζ52 | ζ4ζ54+ζ4ζ5 | ζ4ζ53+ζ4ζ52 | complex lifted from C4×D5 |
| ρ17 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -1 | -1 | -1 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from F5 |
| ρ18 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -1 | -1 | -1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×F5 |
| ρ19 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2+2√5 | -2-2√5 | -2 | 1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ20 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2-2√5 | -2+2√5 | -2 | 1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 20 points - transitive group 20T51
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)
(1 2 3 4 5)(6 10 9 8 7)(11 13 15 12 14)(16 19 17 20 18)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16), (1,2,3,4,5)(6,10,9,8,7)(11,13,15,12,14)(16,19,17,20,18), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16), (1,2,3,4,5)(6,10,9,8,7)(11,13,15,12,14)(16,19,17,20,18), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16)], [(1,2,3,4,5),(6,10,9,8,7),(11,13,15,12,14),(16,19,17,20,18)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15)]])
G:=TransitiveGroup(20,51);
►On 25 points - transitive group 25T18
Generators in S25
(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)
(1 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 23)(24 25)
(1 19 9 14 25)(2 20 10 15 21)(3 16 6 11 22)(4 17 7 12 23)(5 18 8 13 24)
(6 22 11 16)(7 23 12 17)(8 24 13 18)(9 25 14 19)(10 21 15 20)
G:=sub<Sym(25)| (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,23)(24,25), (1,19,9,14,25)(2,20,10,15,21)(3,16,6,11,22)(4,17,7,12,23)(5,18,8,13,24), (6,22,11,16)(7,23,12,17)(8,24,13,18)(9,25,14,19)(10,21,15,20)>;
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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,23)(24,25), (1,19,9,14,25)(2,20,10,15,21)(3,16,6,11,22)(4,17,7,12,23)(5,18,8,13,24), (6,22,11,16)(7,23,12,17)(8,24,13,18)(9,25,14,19)(10,21,15,20) );
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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(21,23),(24,25)], [(1,19,9,14,25),(2,20,10,15,21),(3,16,6,11,22),(4,17,7,12,23),(5,18,8,13,24)], [(6,22,11,16),(7,23,12,17),(8,24,13,18),(9,25,14,19),(10,21,15,20)]])
G:=TransitiveGroup(25,18);
D5×F5 is a maximal quotient of D5.D20 D5.Dic10 Dic5.4F5 D10.F5 Dic5.F5
Matrix representation of D5×F5 ►in GL6(𝔽41)
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 34 | 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 |
| 40 | 7 | 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 | 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 | 40 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,1,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,40,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0] >;
D5×F5 in GAP, Magma, Sage, TeX
D_5\times F_5
% in TeX
G:=Group("D5xF5"); // GroupNames label
G:=SmallGroup(200,41);
// by ID
G=gap.SmallGroup(200,41);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,20,328,2004,1014]);
// Polycyclic
G:=Group<a,b,c,d|a^5=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 D5×F5 in TeX
Character table of D5×F5 in TeX