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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(D4xD5), C22:2D52, C5:D4:2D5, C52:7(C2xD4), (C2xC10):2D10, C5:D20:6C2, (D5xC10):4C22, Dic5:2D5:3C2, (C5xC10).18C23, (C5xDic5):2C22, C10.18(C22xD5), (C2xD52):4C2, C2.18(C2xD52), (C5xC5:D4):4C2, (C2xC5:D5):4C22, (C22xC5:D5):2C2, SmallGroup(400,180)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5xC10 — D10:D10
Chief series
C1 — C5 — C52 — C5xC10 — D5xC10 — C2xD52 — D10:D10
Lower central
C52 — C5xC10 — 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, C2xC4, D4, C23, D5, C10, C10, C2xD4, Dic5, C20, D10, D10, C2xC10, C2xC10, C52, C4xD5, D20, C5:D4, C5:D4, C5xD4, C22xD5, C5xD5, C5:D5, C5:D5, C5xC10, C5xC10, D4xD5, C5xDic5, D52, D5xC10, C2xC5:D5, C2xC5:D5, C102, Dic5:2D5, C5:D20, C5xC5:D4, C2xD52, C22xC5:D5, D10:D10
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, D10, C22xD5, D4xD5, D52, C2xD52, 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+++++++++++++++
imageC1C2C2C2C2C2D4D5D10D10D10D4xD5D52C2xD52D10:D10
kernelD10:D10Dic5:2D5C5:D20C5xC5:D4C2xD52C22xC5:D5C5:D5C5:D4Dic5D10C2xC10C5C22C2C1
# 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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