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G = C10×Dic5order 200 = 23·52

Direct product of C10 and Dic5

direct product, metacyclic, supersoluble, monomial, A-group

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

C1C5 — C10×Dic5
C1C5C10C5×C10C5×Dic5 — C10×Dic5
C5 — C10×Dic5
C1C2×C10

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

2C5
2C5
5C4
5C4
2C10
2C10
2C10
2C10
2C10
2C10
5C2×C4
2C2×C10
2C2×C10
5C20
5C20
5C2×C20

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

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

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

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

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 2A2B2C4A4B4C4D5A5B5C5D5E···5N10A···10L10M···10AP20A···20P
order1222444455555···510···1010···1020···20
size1111555511112···21···12···25···5

80 irreducible representations

dim11111111222222
type++++-+
imageC1C2C2C4C5C10C10C20D5Dic5D10C5×D5C5×Dic5D5×C10
kernelC10×Dic5C5×Dic5C102C5×C10C2×Dic5Dic5C2×C10C10C2×C10C10C10C22C2C2
# reps1214484162428168

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

3100
0180
0018
,
100
0250
0023
,
100
001
0400
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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Subgroup lattice of C10×Dic5 in TeX

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