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## G = D5×D20order 400 = 24·52

### Direct product of D5 and D20

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 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 10M 10N 10O 10P 20A 20B 20C 20D 20E ··· 20N 20O 20P 20Q 20R order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 ··· 20 20 20 20 20 size 1 1 5 5 10 10 50 50 2 10 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 10 10 10 10 20 20 20 20 2 2 2 2 4 ··· 4 10 10 10 10

52 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D5 D5 D10 D10 D10 D20 D4×D5 D52 C2×D52 D5×D20 kernel D5×D20 C5⋊D20 D5×C20 C5×D20 C20⋊D5 C2×D52 C5×D5 C4×D5 D20 Dic5 C20 D10 D5 C5 C4 C2 C1 # reps 1 2 1 1 1 2 2 2 2 2 4 6 8 2 4 4 8

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

 1 0 0 0 0 1 0 0 0 0 6 40 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 6 40 0 0 35 35
,
 13 39 0 0 2 25 0 0 0 0 1 0 0 0 0 1
,
 39 16 0 0 28 2 0 0 0 0 1 0 0 0 0 1
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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