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G = D5×D20  order 400 = 24·52

Direct product of D5 and D20

direct product, metabelian, supersoluble, monomial

Aliases: D5×D20, C20⋊1D10, D10⋊1D10, Dic5⋊3D10, C4⋊1D52, C5⋊1(D4×D5), (C4×D5)⋊3D5, (C5×D5)⋊1D4, C5⋊1(C2×D20), C52⋊2(C2×D4), (C5×D20)⋊6C2, (D5×C20)⋊4C2, C5⋊D20⋊1C2, C20⋊D5⋊5C2, (C5×C20)⋊1C22, (D5×C10)⋊1C22, (C5×C10).8C23, C10.8(C22×D5), (C5×Dic5)⋊3C22, (C2×D52)⋊1C2, C2.10(C2×D52), (C2×C5⋊D5)⋊1C22, SmallGroup(400,170)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5×C10 — D5×D20
Chief series
C1 — C5 — C52 — C5×C10 — D5×C10 — C2×D52 — D5×D20
Lower central
C52 — C5×C10 — D5×D20
Upper central
C1 — C2 — C4

Generators and relations for D5×D20
 G = < a,b,c,d | a5=b2=c20=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 1028 in 124 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D4, C23, D5, D5, C10, C10, C2×D4, Dic5, C20, C20, D10, D10, D10, C2×C10, C52, C4×D5, D20, D20, C5⋊D4, C2×C20, C5×D4, C22×D5, C5×D5, C5×D5, C5⋊D5, C5×C10, C2×D20, D4×D5, C5×Dic5, C5×C20, D52, D5×C10, D5×C10, C2×C5⋊D5, C5⋊D20, D5×C20, C5×D20, C20⋊D5, C2×D52, D5×D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, D20, C22×D5, C2×D20, D4×D5, D52, C2×D52, D5×D20

Smallest permutation representation of D5×D20
►On 40 points
Generators in S40
(1 13 5 17 9)(2 14 6 18 10)(3 15 7 19 11)(4 16 8 20 12)(21 29 37 25 33)(22 30 38 26 34)(23 31 39 27 35)(24 32 40 28 36)

(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 21)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)

(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 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)

(1 25)(2 24)(3 23)(4 22)(5 21)(6 40)(7 39)(8 38)(9 37)(10 36)(11 35)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(18 28)(19 27)(20 26)
 
G:=sub<Sym(40)| (1,13,5,17,9)(2,14,6,18,10)(3,15,7,19,11)(4,16,8,20,12)(21,29,37,25,33)(22,30,38,26,34)(23,31,39,27,35)(24,32,40,28,36), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,25)(2,24)(3,23)(4,22)(5,21)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)>;
 
G:=Group( (1,13,5,17,9)(2,14,6,18,10)(3,15,7,19,11)(4,16,8,20,12)(21,29,37,25,33)(22,30,38,26,34)(23,31,39,27,35)(24,32,40,28,36), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40), (1,25)(2,24)(3,23)(4,22)(5,21)(6,40)(7,39)(8,38)(9,37)(10,36)(11,35)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26) );
 
G=PermutationGroup([[(1,13,5,17,9),(2,14,6,18,10),(3,15,7,19,11),(4,16,8,20,12),(21,29,37,25,33),(22,30,38,26,34),(23,31,39,27,35),(24,32,40,28,36)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,21),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,40),(7,39),(8,38),(9,37),(10,36),(11,35),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(18,28),(19,27),(20,26)]])
 

52 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B5C5D5E5F5G5H10A10B10C10D10E10F10G10H10I10J10K10L10M10N10O10P20A20B20C20D20E···20N20O20P20Q20R
order122222224455555555101010101010101010101010101010102020202020···2020202020
size1155101050502102222444422224444101010102020202022224···410101010

52 irreducible representations

dim11111122222224444
type+++++++++++++++++
imageC1C2C2C2C2C2D4D5D5D10D10D10D20D4×D5D52C2×D52D5×D20
kernelD5×D20C5⋊D20D5×C20C5×D20C20⋊D5C2×D52C5×D5C4×D5D20Dic5C20D10D5C5C4C2C1
# reps12111222224682448

Matrix representation of D5×D20 ►in GL4(𝔽41) generated by

1000
0100
00640
0010
,
1000
0100
00640
003535
,
133900
22500
0010
0001
,
391600
28200
0010
0001
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,6,35,0,0,40,35],[13,2,0,0,39,25,0,0,0,0,1,0,0,0,0,1],[39,28,0,0,16,2,0,0,0,0,1,0,0,0,0,1] >;
 

D5×D20 in GAP, Magma, Sage, TeX 

D_5\times D_{20}
 
% in TeX
 
G:=Group("D5xD20");
 
// GroupNames label
 
G:=SmallGroup(400,170);
 
// by ID
 
G=gap.SmallGroup(400,170);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,116,50,970,11525]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^2=c^20=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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