metabelian, supersoluble, monomial, A-group
Aliases: D5⋊F5, D52.2C2, C5⋊F5⋊C2, C5⋊1(C2×F5), (C5×D5)⋊2C4, C52⋊2(C2×C4), C52⋊C4⋊1C2, C5⋊D5.1C22, Hol(D5), SmallGroup(200,42)
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, dad-1=a3, bc=cb, dbd-1=a2b, dcd-1=c3 >
5C2
5C2
25C2
2C5
2C5
25C4
25C22
25C4
5D5
5C10
5C10
5D5
10D5
10D5
25C2×C4
5F5
5F5
5F5
5D10
5F5
5D10
10F5
10F5
Character table of D5⋊F5
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 10A | 10B | |
| size | 1 | 5 | 5 | 25 | 25 | 25 | 25 | 25 | 4 | 4 | 8 | 8 | 20 | 20 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | -1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | -1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | -1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | -1 | linear of order 4 |
| ρ9 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | -1 | orthogonal lifted from F5 |
| ρ10 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | 0 | 1 | orthogonal lifted from C2×F5 |
| ρ11 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | 1 | 0 | orthogonal lifted from C2×F5 |
| ρ12 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | 0 | orthogonal lifted from F5 |
| ρ13 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 3 | -2 | 0 | 0 | orthogonal faithful |
| ρ14 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 3 | 0 | 0 | orthogonal faithful |
►On 10 points - transitive group 10T17
Generators in S10
(1 2 3 4 5)(6 7 8 9 10)
(1 8)(2 7)(3 6)(4 10)(5 9)
(1 2 3 4 5)(6 10 9 8 7)
(2 3 5 4)(6 8 7 10)
G:=sub<Sym(10)| (1,2,3,4,5)(6,7,8,9,10), (1,8)(2,7)(3,6)(4,10)(5,9), (1,2,3,4,5)(6,10,9,8,7), (2,3,5,4)(6,8,7,10)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,8)(2,7)(3,6)(4,10)(5,9), (1,2,3,4,5)(6,10,9,8,7), (2,3,5,4)(6,8,7,10) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,8),(2,7),(3,6),(4,10),(5,9)], [(1,2,3,4,5),(6,10,9,8,7)], [(2,3,5,4),(6,8,7,10)]])
G:=TransitiveGroup(10,17);
►On 20 points - transitive group 20T54
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 8)(2 7)(3 6)(4 10)(5 9)(11 18)(12 17)(13 16)(14 20)(15 19)
(1 2 3 4 5)(6 10 9 8 7)(11 12 13 14 15)(16 20 19 18 17)
(1 12)(2 14 5 15)(3 11 4 13)(6 17 7 19)(8 16 10 20)(9 18)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,8)(2,7)(3,6)(4,10)(5,9)(11,18)(12,17)(13,16)(14,20)(15,19), (1,2,3,4,5)(6,10,9,8,7)(11,12,13,14,15)(16,20,19,18,17), (1,12)(2,14,5,15)(3,11,4,13)(6,17,7,19)(8,16,10,20)(9,18)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,8)(2,7)(3,6)(4,10)(5,9)(11,18)(12,17)(13,16)(14,20)(15,19), (1,2,3,4,5)(6,10,9,8,7)(11,12,13,14,15)(16,20,19,18,17), (1,12)(2,14,5,15)(3,11,4,13)(6,17,7,19)(8,16,10,20)(9,18) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,8),(2,7),(3,6),(4,10),(5,9),(11,18),(12,17),(13,16),(14,20),(15,19)], [(1,2,3,4,5),(6,10,9,8,7),(11,12,13,14,15),(16,20,19,18,17)], [(1,12),(2,14,5,15),(3,11,4,13),(6,17,7,19),(8,16,10,20),(9,18)]])
G:=TransitiveGroup(20,54);
►On 25 points - transitive group 25T19
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)
(2 3 5 4)(6 13 23 20)(7 15 22 18)(8 12 21 16)(9 14 25 19)(10 11 24 17)
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), (2,3,5,4)(6,13,23,20)(7,15,22,18)(8,12,21,16)(9,14,25,19)(10,11,24,17)>;
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), (2,3,5,4)(6,13,23,20)(7,15,22,18)(8,12,21,16)(9,14,25,19)(10,11,24,17) );
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)], [(2,3,5,4),(6,13,23,20),(7,15,22,18),(8,12,21,16),(9,14,25,19),(10,11,24,17)]])
G:=TransitiveGroup(25,19);
D5⋊F5 is a maximal subgroup of
F52 C52⋊M4(2) D5≀C2⋊C2
D5⋊F5 is a maximal quotient of C52⋊3C42 D10⋊F5 Dic5⋊F5 D10.2F5 C52⋊4M4(2)
| action | f(x) | Disc(f) |
|---|---|---|
| 10T17 | x10-40x8+515x6-48x5-2600x4+390x3+4225x2-507 | 214·35·510·114·138·9438192 |
Matrix representation of D5⋊F5 ►in GL8(ℤ)
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1],[0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0],[-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0],[1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1] >;
D5⋊F5 in GAP, Magma, Sage, TeX
D_5\rtimes F_5
% in TeX
G:=Group("D5:F5"); // GroupNames label
G:=SmallGroup(200,42);
// by ID
G=gap.SmallGroup(200,42);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-5,20,483,328,173,3004,1014]);
// 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,d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of D5⋊F5 in TeX
Character table of D5⋊F5 in TeX