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G = D10⋊D10  order 400 = 24·52

3rd semidirect product of D10 and D10 acting via D10/D5=C2

metabelian, supersoluble, monomial

Aliases: D10⋊3D10, Dic5⋊2D10, C102⋊2C22, C5⋊D5⋊3D4, C5⋊3(D4×D5), C22⋊2D52, C5⋊D4⋊2D5, C52⋊7(C2×D4), (C2×C10)⋊2D10, C5⋊D20⋊6C2, (D5×C10)⋊4C22, Dic5⋊2D5⋊3C2, (C5×C10).18C23, (C5×Dic5)⋊2C22, C10.18(C22×D5), (C2×D52)⋊4C2, C2.18(C2×D52), (C5×C5⋊D4)⋊4C2, (C2×C5⋊D5)⋊4C22, (C22×C5⋊D5)⋊2C2, SmallGroup(400,180)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C10 — D10⋊D10
Chief series
C1 — C5 — C52 — C5×C10 — D5×C10 — C2×D52 — D10⋊D10
Lower central
C52 — C5×C10 — D10⋊D10
Upper central
C1 — C2 — C22

Generators and relations for D10⋊D10
 G = < a,b,c,d | a10=b2=c10=d2=1, bab=dad=a-1, ac=ca, cbc-1=a5b, dbd=a3b, dcd=c-1 >

Subgroups: 1100 in 140 conjugacy classes, 34 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C5, C5, C2×C4, D4, C23, D5, C10, C10, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C52, C4×D5, D20, C5⋊D4, C5⋊D4, C5×D4, C22×D5, C5×D5, C5⋊D5, C5⋊D5, C5×C10, C5×C10, D4×D5, C5×Dic5, D52, D5×C10, C2×C5⋊D5, C2×C5⋊D5, C102, Dic5⋊2D5, C5⋊D20, C5×C5⋊D4, C2×D52, C22×C5⋊D5, D10⋊D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22×D5, D4×D5, D52, C2×D52, D10⋊D10

Permutation representations of D10⋊D10
►On 20 points - transitive group 20T106
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

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

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

(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 20)(16 19)(17 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), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13), (1,5,9,3,7)(2,6,10,4,8)(11,12,13,14,15,16,17,18,19,20), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,20)(16,19)(17,18)>;
 
G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12)(2,11)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,13), (1,5,9,3,7)(2,6,10,4,8)(11,12,13,14,15,16,17,18,19,20), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,20)(16,19)(17,18) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,12),(2,11),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,13)], [(1,5,9,3,7),(2,6,10,4,8),(11,12,13,14,15,16,17,18,19,20)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,20),(16,19),(17,18)]])
 
G:=TransitiveGroup(20,106);
 

46 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B5C5D5E5F5G5H10A10B10C10D10E···10T10U10V10W10X20A20B20C20D
order1222222244555555551010101010···101010101020202020
size112101025255010102222444422224···42020202020202020

46 irreducible representations

dim111111222224444
type+++++++++++++++
imageC1C2C2C2C2C2D4D5D10D10D10D4×D5D52C2×D52D10⋊D10
kernelD10⋊D10Dic5⋊2D5C5⋊D20C5×C5⋊D4C2×D52C22×C5⋊D5C5⋊D5C5⋊D4Dic5D10C2×C10C5C22C2C1
# reps112211244444448

Matrix representation of D10⋊D10 ►in GL6(𝔽41)

4000000
0400000
0035600
00354000
000010
000001
,
120000
0400000
00354000
0035600
000010
000001
,
100000
40400000
001000
000100
0000734
0000740
,
100000
40400000
000100
001000
0000740
0000734

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,35,0,0,0,0,6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,2,40,0,0,0,0,0,0,35,35,0,0,0,0,40,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,34,40],[1,40,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,7,0,0,0,0,40,34] >;
 

D10⋊D10 in GAP, Magma, Sage, TeX 

D_{10}\rtimes D_{10}
 
% in TeX
 
G:=Group("D10:D10");
 
// GroupNames label
 
G:=SmallGroup(400,180);
 
// by ID
 
G=gap.SmallGroup(400,180);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,218,116,970,11525]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^10=b^2=c^10=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^5*b,d*b*d=a^3*b,d*c*d=c^-1>;
 
// generators/relations
 

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