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## G = C10×Dic5order 200 = 23·52

### Direct product of C10 and Dic5

Aliases: C10×Dic5, C102C20, C10.20D10, C102.1C2, (C5×C10)⋊5C4, C53(C2×C20), C22.(C5×D5), C5211(C2×C4), C2.2(D5×C10), (C2×C10).5D5, (C2×C10).2C10, C10.4(C2×C10), (C5×C10).9C22, SmallGroup(200,30)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C10×Dic5
 Chief series C1 — C5 — C10 — C5×C10 — C5×Dic5 — C10×Dic5
 Lower central C5 — C10×Dic5
 Upper central C1 — C2×C10

Generators and relations for C10×Dic5
G = < a,b,c | a10=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C10×Dic5
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 21 9 29 7 27 5 25 3 23)(2 22 10 30 8 28 6 26 4 24)(11 38 13 40 15 32 17 34 19 36)(12 39 14 31 16 33 18 35 20 37)
(1 18 27 39)(2 19 28 40)(3 20 29 31)(4 11 30 32)(5 12 21 33)(6 13 22 34)(7 14 23 35)(8 15 24 36)(9 16 25 37)(10 17 26 38)

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

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

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

C10×Dic5 is a maximal subgroup of   D10⋊Dic5  C10.D20  Dic5⋊Dic5  C10.Dic10  C102.C4  Dic5.D10  D5×C2×C20

80 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E ··· 5N 10A ··· 10L 10M ··· 10AP 20A ··· 20P order 1 2 2 2 4 4 4 4 5 5 5 5 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 5 5 5 5 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C4 C5 C10 C10 C20 D5 Dic5 D10 C5×D5 C5×Dic5 D5×C10 kernel C10×Dic5 C5×Dic5 C102 C5×C10 C2×Dic5 Dic5 C2×C10 C10 C2×C10 C10 C10 C22 C2 C2 # reps 1 2 1 4 4 8 4 16 2 4 2 8 16 8

Matrix representation of C10×Dic5 in GL3(𝔽41) generated by

 31 0 0 0 18 0 0 0 18
,
 1 0 0 0 25 0 0 0 23
,
 1 0 0 0 0 1 0 40 0
G:=sub<GL(3,GF(41))| [31,0,0,0,18,0,0,0,18],[1,0,0,0,25,0,0,0,23],[1,0,0,0,0,40,0,1,0] >;

C10×Dic5 in GAP, Magma, Sage, TeX

C_{10}\times {\rm Dic}_5
% in TeX

G:=Group("C10xDic5");
// GroupNames label

G:=SmallGroup(200,30);
// by ID

G=gap.SmallGroup(200,30);
# by ID

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

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

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