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## G = D5.D5order 100 = 22·52

### The non-split extension by D5 of D5 acting via D5/C5=C2

Aliases: D5.D5, C5⋊Dic5, C53F5, C522C4, (C5×D5).2C2, SmallGroup(100,10)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5.D5
G = < a,b,c,d | a5=b2=c5=1, d2=a-1b, bab=a-1, ac=ca, dad-1=a2, bc=cb, dbd-1=ab, dcd-1=c-1 >

Character table of D5.D5

 class 1 2 4A 4B 5A 5B 5C 5D 5E 5F 5G 10A 10B size 1 5 25 25 2 2 4 4 4 4 4 10 10 ρ1 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 linear of order 2 ρ3 1 -1 -i i 1 1 1 1 1 1 1 -1 -1 linear of order 4 ρ4 1 -1 i -i 1 1 1 1 1 1 1 -1 -1 linear of order 4 ρ5 2 2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ6 2 2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ7 2 -2 0 0 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 2 -1+√5/2 -1-√5/2 1+√5/2 1-√5/2 symplectic lifted from Dic5, Schur index 2 ρ8 2 -2 0 0 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 2 -1-√5/2 -1+√5/2 1-√5/2 1+√5/2 symplectic lifted from Dic5, Schur index 2 ρ9 4 0 0 0 4 4 -1 -1 -1 -1 -1 0 0 orthogonal lifted from F5 ρ10 4 0 0 0 -1+√5 -1-√5 2ζ52+ζ5+1 ζ53+2ζ5+1 -1 2ζ54+ζ52+1 ζ54+2ζ53+1 0 0 complex faithful ρ11 4 0 0 0 -1+√5 -1-√5 ζ54+2ζ53+1 2ζ54+ζ52+1 -1 ζ53+2ζ5+1 2ζ52+ζ5+1 0 0 complex faithful ρ12 4 0 0 0 -1-√5 -1+√5 2ζ54+ζ52+1 2ζ52+ζ5+1 -1 ζ54+2ζ53+1 ζ53+2ζ5+1 0 0 complex faithful ρ13 4 0 0 0 -1-√5 -1+√5 ζ53+2ζ5+1 ζ54+2ζ53+1 -1 2ζ52+ζ5+1 2ζ54+ζ52+1 0 0 complex faithful

Permutation representations of D5.D5
On 20 points - transitive group 20T26
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 18)(12 17)(13 16)(14 20)(15 19)
(1 5 4 3 2)(6 7 8 9 10)(11 13 15 12 14)(16 19 17 20 18)
(1 19 6 11)(2 17 10 13)(3 20 9 15)(4 18 8 12)(5 16 7 14)

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

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

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

G:=TransitiveGroup(20,26);

On 25 points - transitive group 25T11
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)(22 25)(23 24)
(1 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)
(1 9)(2 7 5 6)(3 10 4 8)(11 25 12 23)(13 21 15 22)(14 24)(16 20 17 18)

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

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

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

G:=TransitiveGroup(25,11);

D5.D5 is a maximal subgroup of
D5×F5  D5.D15  D5.D25  D25.D5  He5⋊C4  C52⋊F5  C53⋊C4  C536C4  C537C4
D5.D5 is a maximal quotient of
C523C8  D5.D15  D5.D25  D25.D5  He5⋊C4  C53⋊C4  C536C4  C537C4

Matrix representation of D5.D5 in GL4(𝔽41) generated by

 18 0 0 0 0 16 0 0 0 0 10 0 0 0 0 37
,
 0 16 0 0 18 0 0 0 0 0 0 37 0 0 10 0
,
 16 0 0 0 0 16 0 0 0 0 18 0 0 0 0 18
,
 0 0 18 0 0 0 0 18 0 16 0 0 16 0 0 0
G:=sub<GL(4,GF(41))| [18,0,0,0,0,16,0,0,0,0,10,0,0,0,0,37],[0,18,0,0,16,0,0,0,0,0,0,10,0,0,37,0],[16,0,0,0,0,16,0,0,0,0,18,0,0,0,0,18],[0,0,0,16,0,0,16,0,18,0,0,0,0,18,0,0] >;

D5.D5 in GAP, Magma, Sage, TeX

D_5.D_5
% in TeX

G:=Group("D5.D5");
// GroupNames label

G:=SmallGroup(100,10);
// by ID

G=gap.SmallGroup(100,10);
# by ID

G:=PCGroup([4,-2,-2,-5,-5,8,194,963,647]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^5=1,d^2=a^-1*b,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^2,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

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