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## G = C5×D20order 200 = 23·52

### Direct product of C5 and D20

Aliases: C5×D20, C203D5, C201C10, C524D4, D101C10, C10.19D10, C4⋊(C5×D5), C51(C5×D4), (C5×C20)⋊2C2, (D5×C10)⋊3C2, C2.4(D5×C10), C10.3(C2×C10), (C5×C10).8C22, SmallGroup(200,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×D20
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — C5×D20
 Lower central C5 — C10 — C5×D20
 Upper central C1 — C10 — C20

Generators and relations for C5×D20
G = < a,b,c | a5=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C5×D20
On 40 points
Generators in S40
(1 5 9 13 17)(2 6 10 14 18)(3 7 11 15 19)(4 8 12 16 20)(21 37 33 29 25)(22 38 34 30 26)(23 39 35 31 27)(24 40 36 32 28)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 40)(17 39)(18 38)(19 37)(20 36)

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

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

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

C5×D20 is a maximal subgroup of
C522D8  C5⋊D40  D20.D5  C523SD16  D205D5  D20⋊D5  C20⋊D10  C5×D4×D5

65 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10N 10O ··· 10V 20A ··· 20X order 1 2 2 2 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 10 10 2 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 10 ··· 10 2 ··· 2

65 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 D4 D5 D10 D20 C5×D4 C5×D5 D5×C10 C5×D20 kernel C5×D20 C5×C20 D5×C10 D20 C20 D10 C52 C20 C10 C5 C5 C4 C2 C1 # reps 1 1 2 4 4 8 1 2 2 4 4 8 8 16

Matrix representation of C5×D20 in GL2(𝔽41) generated by

 10 0 0 10
,
 36 0 0 8
,
 0 8 36 0
G:=sub<GL(2,GF(41))| [10,0,0,10],[36,0,0,8],[0,36,8,0] >;

C5×D20 in GAP, Magma, Sage, TeX

C_5\times D_{20}
% in TeX

G:=Group("C5xD20");
// GroupNames label

G:=SmallGroup(200,29);
// by ID

G=gap.SmallGroup(200,29);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-5,221,106,4004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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