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G = C5×C52C8order 200 = 23·52

Direct product of C5 and C52C8

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

Aliases: C5×C52C8, C52C40, C526C8, C20.8D5, C20.2C10, C10.2C20, C10.5Dic5, C4.2(C5×D5), C2.(C5×Dic5), (C5×C10).5C4, (C5×C20).3C2, SmallGroup(200,15)

Series: Derived Chief Lower central Upper central

C1C5 — C5×C52C8
C1C5C10C20C5×C20 — C5×C52C8
C5 — C5×C52C8
C1C20

Generators and relations for C5×C52C8
 G = < a,b,c | a5=b5=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

2C5
2C5
2C10
2C10
5C8
2C20
2C20
5C40

Smallest permutation representation of C5×C52C8
On 40 points
Generators in S40
(1 11 20 27 35)(2 12 21 28 36)(3 13 22 29 37)(4 14 23 30 38)(5 15 24 31 39)(6 16 17 32 40)(7 9 18 25 33)(8 10 19 26 34)
(1 11 20 27 35)(2 36 28 21 12)(3 13 22 29 37)(4 38 30 23 14)(5 15 24 31 39)(6 40 32 17 16)(7 9 18 25 33)(8 34 26 19 10)
(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)

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

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

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

C5×C52C8 is a maximal subgroup of
C523C16  C20.29D10  C20.30D10  C20.31D10  C5⋊D40  C523SD16  C524SD16  C523Q16  D5×C40

80 conjugacy classes

class 1  2 4A4B5A5B5C5D5E···5N8A8B8C8D10A10B10C10D10E···10N20A···20H20I···20AB40A···40P
order124455555···588881010101010···1020···2020···2040···40
size111111112···2555511112···21···12···25···5

80 irreducible representations

dim11111111222222
type+++-
imageC1C2C4C5C8C10C20C40D5Dic5C52C8C5×D5C5×Dic5C5×C52C8
kernelC5×C52C8C5×C20C5×C10C52C8C52C20C10C5C20C10C5C4C2C1
# reps1124448162248816

Matrix representation of C5×C52C8 in GL2(𝔽41) generated by

160
016
,
160
018
,
01
320
G:=sub<GL(2,GF(41))| [16,0,0,16],[16,0,0,18],[0,32,1,0] >;

C5×C52C8 in GAP, Magma, Sage, TeX

C_5\times C_5\rtimes_2C_8
% in TeX

G:=Group("C5xC5:2C8");
// GroupNames label

G:=SmallGroup(200,15);
// by ID

G=gap.SmallGroup(200,15);
# by ID

G:=PCGroup([5,-2,-5,-2,-2,-5,50,42,4004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×C52C8 in TeX

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