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## G = D5×C20order 200 = 23·52

### Direct product of C20 and D5

Aliases: D5×C20, C202C10, Dic52C10, D10.2C10, C10.18D10, C52(C2×C20), (C5×C20)⋊3C2, C529(C2×C4), C2.1(D5×C10), C10.2(C2×C10), (C5×Dic5)⋊5C2, (D5×C10).4C2, (C5×C10).7C22, SmallGroup(200,28)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×C20
G = < a,b,c | a20=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C20
On 40 points
Generators in S40
(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 17 13 9 5)(2 18 14 10 6)(3 19 15 11 7)(4 20 16 12 8)(21 25 29 33 37)(22 26 30 34 38)(23 27 31 35 39)(24 28 32 36 40)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)

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

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

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

D5×C20 is a maximal subgroup of   C20.30D10  C20.14F5  C20.12F5  C205F5  D205D5  D10.9D10  Dic105D5

80 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10N 10O ··· 10V 20A ··· 20H 20I ··· 20AB 20AC ··· 20AJ order 1 2 2 2 4 4 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 5 5 1 1 5 5 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 5 ··· 5 1 ··· 1 2 ··· 2 5 ··· 5

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 D5 D10 C4×D5 C5×D5 D5×C10 D5×C20 kernel D5×C20 C5×Dic5 C5×C20 D5×C10 C5×D5 C4×D5 Dic5 C20 D10 D5 C20 C10 C5 C4 C2 C1 # reps 1 1 1 1 4 4 4 4 4 16 2 2 4 8 8 16

Matrix representation of D5×C20 in GL2(𝔽41) generated by

 2 0 0 2
,
 18 0 0 16
,
 0 16 18 0
G:=sub<GL(2,GF(41))| [2,0,0,2],[18,0,0,16],[0,18,16,0] >;

D5×C20 in GAP, Magma, Sage, TeX

D_5\times C_{20}
% in TeX

G:=Group("D5xC20");
// GroupNames label

G:=SmallGroup(200,28);
// by ID

G=gap.SmallGroup(200,28);
# by ID

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

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

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