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## G = C5×Dic10order 200 = 23·52

### Direct product of C5 and Dic10

Aliases: C5×Dic10, C523Q8, C20.7D5, C20.1C10, C10.17D10, Dic5.1C10, C5⋊(C5×Q8), C4.(C5×D5), (C5×C20).2C2, C2.3(D5×C10), C10.1(C2×C10), (C5×C10).6C22, (C5×Dic5).4C2, SmallGroup(200,27)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×Dic10
 Chief series C1 — C5 — C10 — C5×C10 — C5×Dic5 — C5×Dic10
 Lower central C5 — C10 — C5×Dic10
 Upper central C1 — C10 — C20

Generators and relations for C5×Dic10
G = < a,b,c | a5=b20=1, c2=b10, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5×Dic10
On 40 points
Generators in S40
(1 9 17 5 13)(2 10 18 6 14)(3 11 19 7 15)(4 12 20 8 16)(21 33 25 37 29)(22 34 26 38 30)(23 35 27 39 31)(24 36 28 40 32)
(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 31 11 21)(2 30 12 40)(3 29 13 39)(4 28 14 38)(5 27 15 37)(6 26 16 36)(7 25 17 35)(8 24 18 34)(9 23 19 33)(10 22 20 32)

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

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

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

C5×Dic10 is a maximal subgroup of
D20.D5  C524SD16  C522Q16  C523Q16  D20⋊D5  Dic10⋊D5  Dic105D5  C5×Q8×D5

65 conjugacy classes

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

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 Q8 D5 D10 Dic10 C5×Q8 C5×D5 D5×C10 C5×Dic10 kernel C5×Dic10 C5×Dic5 C5×C20 Dic10 Dic5 C20 C52 C20 C10 C5 C5 C4 C2 C1 # reps 1 2 1 4 8 4 1 2 2 4 4 8 8 16

Matrix representation of C5×Dic10 in GL2(𝔽41) generated by

 10 0 0 10
,
 2 0 0 21
,
 0 1 40 0
G:=sub<GL(2,GF(41))| [10,0,0,10],[2,0,0,21],[0,40,1,0] >;

C5×Dic10 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{10}
% in TeX

G:=Group("C5xDic10");
// GroupNames label

G:=SmallGroup(200,27);
// by ID

G=gap.SmallGroup(200,27);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-5,100,221,106,4004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^20=1,c^2=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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