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## G = C5×C5⋊2C8order 200 = 23·52

### Direct product of C5 and C5⋊2C8

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

 Derived series C1 — C5 — C5×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C5×C20 — C5×C5⋊2C8
 Lower central C5 — C5×C5⋊2C8
 Upper central C1 — C20

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

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 4A 4B 5A 5B 5C 5D 5E ··· 5N 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10N 20A ··· 20H 20I ··· 20AB 40A ··· 40P order 1 2 4 4 5 5 5 5 5 ··· 5 8 8 8 8 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 1 1 1 1 2 ··· 2 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 C4 C5 C8 C10 C20 C40 D5 Dic5 C5⋊2C8 C5×D5 C5×Dic5 C5×C5⋊2C8 kernel C5×C5⋊2C8 C5×C20 C5×C10 C5⋊2C8 C52 C20 C10 C5 C20 C10 C5 C4 C2 C1 # reps 1 1 2 4 4 4 8 16 2 2 4 8 8 16

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

 16 0 0 16
,
 16 0 0 18
,
 0 1 32 0
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

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