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## G = C10×F5order 200 = 23·52

### Direct product of C10 and F5

Aliases: C10×F5, C10⋊C20, D5⋊C20, D10.C10, C5⋊(C2×C20), (C5×C10)⋊1C4, (C5×D5)⋊3C4, D5.(C2×C10), C523(C2×C4), (D5×C10).2C2, (C5×D5).2C22, SmallGroup(200,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C10×F5
 Chief series C1 — C5 — D5 — C5×D5 — C5×F5 — C10×F5
 Lower central C5 — C10×F5
 Upper central C1 — C10

Generators and relations for C10×F5
G = < a,b,c | a10=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C10×F5
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 3 5 7 9)(2 4 6 8 10)(11 17 13 19 15)(12 18 14 20 16)(21 29 27 25 23)(22 30 28 26 24)(31 35 39 33 37)(32 36 40 34 38)
(1 11 25 32)(2 12 26 33)(3 13 27 34)(4 14 28 35)(5 15 29 36)(6 16 30 37)(7 17 21 38)(8 18 22 39)(9 19 23 40)(10 20 24 31)

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

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

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

C10×F5 is a maximal subgroup of   D5.D20  D5.Dic10

50 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E ··· 5I 10A 10B 10C 10D 10E ··· 10I 10J ··· 10Q 20A ··· 20P order 1 2 2 2 4 4 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 5 5 5 5 5 5 1 1 1 1 4 ··· 4 1 1 1 1 4 ··· 4 5 ··· 5 5 ··· 5

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 type + + + + + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 F5 C2×F5 C5×F5 C10×F5 kernel C10×F5 C5×F5 D5×C10 C5×D5 C5×C10 C2×F5 F5 D10 D5 C10 C10 C5 C2 C1 # reps 1 2 1 2 2 4 8 4 8 8 1 1 4 4

Matrix representation of C10×F5 in GL5(𝔽41)

 25 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 10 0 0 35 0 0 37 0 20 0 0 0 18 18 0 0 0 0 16
,
 40 0 0 0 0 0 32 0 1 0 0 9 0 0 0 0 1 1 0 0 0 9 0 0 9

G:=sub<GL(5,GF(41))| [25,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,10,0,0,0,0,0,37,0,0,0,0,0,18,0,0,35,20,18,16],[40,0,0,0,0,0,32,9,1,9,0,0,0,1,0,0,1,0,0,0,0,0,0,0,9] >;

C10×F5 in GAP, Magma, Sage, TeX

C_{10}\times F_5
% in TeX

G:=Group("C10xF5");
// GroupNames label

G:=SmallGroup(200,45);
// by ID

G=gap.SmallGroup(200,45);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-5,100,2004,219]);
// Polycyclic

G:=Group<a,b,c|a^10=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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