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## G = D5×F5order 200 = 23·52

### Direct product of D5 and F5

Aliases: D5×F5, D5.1D10, C5⋊D5⋊C4, C5⋊(C4×D5), (C5×F5)⋊C2, D52.1C2, D5.D5⋊C2, C54(C2×F5), (C5×D5)⋊1C4, C521(C2×C4), (C5×D5).C22, SmallGroup(200,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D5×F5
 Chief series C1 — C5 — C52 — C5×D5 — C5×F5 — D5×F5
 Lower central C52 — D5×F5
 Upper central C1

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

Permutation representations of D5×F5
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 9 14 19 25)(2 10 15 20 21)(3 6 11 16 22)(4 7 12 17 23)(5 8 13 18 24)
(6 11 22 16)(7 12 23 17)(8 13 24 18)(9 14 25 19)(10 15 21 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), (1,9,14,19,25)(2,10,15,20,21)(3,6,11,16,22)(4,7,12,17,23)(5,8,13,18,24), (6,11,22,16)(7,12,23,17)(8,13,24,18)(9,14,25,19)(10,15,21,20)>;

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), (1,9,14,19,25)(2,10,15,20,21)(3,6,11,16,22)(4,7,12,17,23)(5,8,13,18,24), (6,11,22,16)(7,12,23,17)(8,13,24,18)(9,14,25,19)(10,15,21,20) );

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)], [(1,9,14,19,25),(2,10,15,20,21),(3,6,11,16,22),(4,7,12,17,23),(5,8,13,18,24)], [(6,11,22,16),(7,12,23,17),(8,13,24,18),(9,14,25,19),(10,15,21,20)])

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

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