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G = D5⋊F5order 200 = 23·52

The semidirect product of D5 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, A-group

Aliases: D5⋊F5, D52.2C2, C5⋊F5⋊C2, C51(C2×F5), (C5×D5)⋊2C4, C522(C2×C4), C52⋊C41C2, C5⋊D5.1C22, Hol(D5), SmallGroup(200,42)

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, 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
5C2×F5
5C2×F5

Character table of D5⋊F5

 class 12A2B2C4A4B4C4D5A5B5C5D10A10B
 size 155252525252544882020
ρ111111111111111    trivial
ρ21-1-111-11-11111-1-1    linear of order 2
ρ31-1-11-11-111111-1-1    linear of order 2
ρ41111-1-1-1-1111111    linear of order 2
ρ511-1-1-i-iii1111-11    linear of order 4
ρ611-1-1ii-i-i1111-11    linear of order 4
ρ71-11-1i-i-ii11111-1    linear of order 4
ρ81-11-1-iii-i11111-1    linear of order 4
ρ9440000004-1-1-10-1    orthogonal lifted from F5
ρ104-40000004-1-1-101    orthogonal lifted from C2×F5
ρ1140-400000-14-1-110    orthogonal lifted from C2×F5
ρ1240400000-14-1-1-10    orthogonal lifted from F5
ρ1380000000-2-23-200    orthogonal faithful
ρ1480000000-2-2-2300    orthogonal faithful

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

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

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

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   C523C42  D10⋊F5  Dic5⋊F5  D10.2F5  C524M4(2)

Polynomial with Galois group D5⋊F5 over ℚ
actionf(x)Disc(f)
10T17x10-40x8+515x6-48x5-2600x4+390x3+4225x2-507214·35·510·114·138·9438192

Matrix representation of D5⋊F5 in GL8(ℤ)

-1-1-1-10000
10000000
01000000
00100000
00000100
00000010
00000001
0000-1-1-1-1
,
00000100
00000010
00000001
0000-1-1-1-1
-1-1-1-10000
10000000
01000000
00100000
,
-1-1-1-10000
10000000
01000000
00100000
0000-1-1-1-1
00001000
00000100
00000010
,
10000000
00010000
01000000
-1-1-1-10000
00001000
00000001
00000100
0000-1-1-1-1

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

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