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## G = C5×C5⋊C8order 200 = 23·52

### Direct product of C5 and C5⋊C8

Aliases: C5×C5⋊C8, C5⋊C40, C10.C20, C522C8, C10.6F5, Dic5.2C10, C2.(C5×F5), (C5×C10).1C4, (C5×Dic5).1C2, SmallGroup(200,18)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C5×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C5×Dic5 — C5×C5⋊C8
 Lower central C5 — C5×C5⋊C8
 Upper central C1 — C10

Generators and relations for C5×C5⋊C8
G = < a,b,c | a5=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C5×C5⋊C8
On 40 points
Generators in S40
(1 24 32 9 35)(2 17 25 10 36)(3 18 26 11 37)(4 19 27 12 38)(5 20 28 13 39)(6 21 29 14 40)(7 22 30 15 33)(8 23 31 16 34)
(1 24 32 9 35)(2 10 17 36 25)(3 37 11 26 18)(4 27 38 19 12)(5 20 28 13 39)(6 14 21 40 29)(7 33 15 30 22)(8 31 34 23 16)
(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)| (1,24,32,9,35)(2,17,25,10,36)(3,18,26,11,37)(4,19,27,12,38)(5,20,28,13,39)(6,21,29,14,40)(7,22,30,15,33)(8,23,31,16,34), (1,24,32,9,35)(2,10,17,36,25)(3,37,11,26,18)(4,27,38,19,12)(5,20,28,13,39)(6,14,21,40,29)(7,33,15,30,22)(8,31,34,23,16), (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( (1,24,32,9,35)(2,17,25,10,36)(3,18,26,11,37)(4,19,27,12,38)(5,20,28,13,39)(6,21,29,14,40)(7,22,30,15,33)(8,23,31,16,34), (1,24,32,9,35)(2,10,17,36,25)(3,37,11,26,18)(4,27,38,19,12)(5,20,28,13,39)(6,14,21,40,29)(7,33,15,30,22)(8,31,34,23,16), (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([[(1,24,32,9,35),(2,17,25,10,36),(3,18,26,11,37),(4,19,27,12,38),(5,20,28,13,39),(6,21,29,14,40),(7,22,30,15,33),(8,23,31,16,34)], [(1,24,32,9,35),(2,10,17,36,25),(3,37,11,26,18),(4,27,38,19,12),(5,20,28,13,39),(6,14,21,40,29),(7,33,15,30,22),(8,31,34,23,16)], [(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×C5⋊C8 is a maximal subgroup of   Dic5.4F5  D10.F5  Dic5.F5

50 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E ··· 5I 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10I 20A ··· 20H 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 40 ··· 40 size 1 1 5 5 1 1 1 1 4 ··· 4 5 5 5 5 1 1 1 1 4 ··· 4 5 ··· 5 5 ··· 5

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 4 4 4 4 type + + + - image C1 C2 C4 C5 C8 C10 C20 C40 F5 C5⋊C8 C5×F5 C5×C5⋊C8 kernel C5×C5⋊C8 C5×Dic5 C5×C10 C5⋊C8 C52 Dic5 C10 C5 C10 C5 C2 C1 # reps 1 1 2 4 4 4 8 16 1 1 4 4

Matrix representation of C5×C5⋊C8 in GL5(𝔽41)

 18 0 0 0 0 0 37 0 0 0 0 0 37 0 0 0 0 0 37 0 0 0 0 0 37
,
 1 0 0 0 0 0 37 0 0 37 0 0 10 0 28 0 0 0 16 20 0 0 0 0 18
,
 38 0 0 0 0 0 27 0 1 21 0 3 0 0 16 0 32 1 0 34 0 36 0 0 14

G:=sub<GL(5,GF(41))| [18,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37,0,0,0,0,0,37],[1,0,0,0,0,0,37,0,0,0,0,0,10,0,0,0,0,0,16,0,0,37,28,20,18],[38,0,0,0,0,0,27,3,32,36,0,0,0,1,0,0,1,0,0,0,0,21,16,34,14] >;

C5×C5⋊C8 in GAP, Magma, Sage, TeX

C_5\times C_5\rtimes C_8
% in TeX

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

G:=SmallGroup(200,18);
// by ID

G=gap.SmallGroup(200,18);
# by ID

G:=PCGroup([5,-2,-5,-2,-2,-5,50,42,2004,414]);
// 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^3>;
// generators/relations

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