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