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G = D5×F5order 200 = 23·52

Direct product of D5 and F5

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

Series: Derived Chief Lower central Upper central

C1C52 — D5×F5
C1C5C52C5×D5C5×F5 — D5×F5
C52 — D5×F5
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 >

5C2
5C2
25C2
4C5
5C4
25C22
25C4
5C10
5C10
5D5
5D5
20D5
25C2×C4
5F5
5C20
5Dic5
5D10
5D10
5C4×D5
5C2×F5

Character table of D5×F5

 class 12A2B2C4A4B4C4D5A5B5C5D5E10A10B10C20A20B20C20D
 size 155255525252248810102010101010
ρ111111111111111111111    trivial
ρ21111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ311-1-1-1-1111111111-1-1-1-1-1    linear of order 2
ρ411-1-111-1-11111111-11111    linear of order 2
ρ51-11-1i-i-ii11111-1-11-i-iii    linear of order 4
ρ61-11-1-iii-i11111-1-11ii-i-i    linear of order 4
ρ71-1-11-ii-ii11111-1-1-1ii-i-i    linear of order 4
ρ81-1-11i-ii-i11111-1-1-1-i-iii    linear of order 4
ρ922002200-1+5/2-1-5/22-1+5/2-1-5/2-1-5/2-1+5/20-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ102200-2-200-1-5/2-1+5/22-1-5/2-1+5/2-1+5/2-1-5/201+5/21-5/21+5/21-5/2    orthogonal lifted from D10
ρ1122002200-1-5/2-1+5/22-1-5/2-1+5/2-1+5/2-1-5/20-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ122200-2-200-1+5/2-1-5/22-1+5/2-1-5/2-1-5/2-1+5/201-5/21+5/21-5/21+5/2    orthogonal lifted from D10
ρ132-200-2i2i00-1+5/2-1-5/22-1+5/2-1-5/21+5/21-5/20ζ4ζ544ζ5ζ4ζ534ζ52ζ43ζ5443ζ5ζ43ζ5343ζ52    complex lifted from C4×D5
ρ142-200-2i2i00-1-5/2-1+5/22-1-5/2-1+5/21-5/21+5/20ζ4ζ534ζ52ζ4ζ544ζ5ζ43ζ5343ζ52ζ43ζ5443ζ5    complex lifted from C4×D5
ρ152-2002i-2i00-1-5/2-1+5/22-1-5/2-1+5/21-5/21+5/20ζ43ζ5343ζ52ζ43ζ5443ζ5ζ4ζ534ζ52ζ4ζ544ζ5    complex lifted from C4×D5
ρ162-2002i-2i00-1+5/2-1-5/22-1+5/2-1-5/21+5/21-5/20ζ43ζ5443ζ5ζ43ζ5343ζ52ζ4ζ544ζ5ζ4ζ534ζ52    complex lifted from C4×D5
ρ174040000044-1-1-100-10000    orthogonal lifted from F5
ρ1840-40000044-1-1-10010000    orthogonal lifted from C2×F5
ρ1980000000-2+25-2-25-21-5/21+5/20000000    orthogonal faithful
ρ2080000000-2-25-2+25-21+5/21-5/20000000    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)

0400000
1340000
001000
000100
000010
000001
,
4070000
010000
001000
000100
000010
000001
,
100000
010000
000100
000010
0040404040
001000
,
3200000
0320000
001000
0040404040
000100
000010

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

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