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## G = F5×C25order 500 = 22·53

### Direct product of C25 and F5

Aliases: F5×C25, C5⋊C100, D5.C50, C52.2C20, (C5×C25)⋊1C4, (C5×F5).C5, C5.4(C5×F5), (D5×C25).1C2, (C5×D5).1C10, SmallGroup(500,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — F5×C25
 Chief series C1 — C5 — C52 — C5×D5 — D5×C25 — F5×C25
 Lower central C5 — F5×C25
 Upper central C1 — C25

Generators and relations for F5×C25
G = < a,b,c | a25=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of F5×C25
On 100 points
Generators in S100
(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 41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75)(76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)
(1 6 11 16 21)(2 7 12 17 22)(3 8 13 18 23)(4 9 14 19 24)(5 10 15 20 25)(26 41 31 46 36)(27 42 32 47 37)(28 43 33 48 38)(29 44 34 49 39)(30 45 35 50 40)(51 71 66 61 56)(52 72 67 62 57)(53 73 68 63 58)(54 74 69 64 59)(55 75 70 65 60)(76 86 96 81 91)(77 87 97 82 92)(78 88 98 83 93)(79 89 99 84 94)(80 90 100 85 95)
(1 42 60 77)(2 43 61 78)(3 44 62 79)(4 45 63 80)(5 46 64 81)(6 47 65 82)(7 48 66 83)(8 49 67 84)(9 50 68 85)(10 26 69 86)(11 27 70 87)(12 28 71 88)(13 29 72 89)(14 30 73 90)(15 31 74 91)(16 32 75 92)(17 33 51 93)(18 34 52 94)(19 35 53 95)(20 36 54 96)(21 37 55 97)(22 38 56 98)(23 39 57 99)(24 40 58 100)(25 41 59 76)

G:=sub<Sym(100)| (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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,6,11,16,21)(2,7,12,17,22)(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25)(26,41,31,46,36)(27,42,32,47,37)(28,43,33,48,38)(29,44,34,49,39)(30,45,35,50,40)(51,71,66,61,56)(52,72,67,62,57)(53,73,68,63,58)(54,74,69,64,59)(55,75,70,65,60)(76,86,96,81,91)(77,87,97,82,92)(78,88,98,83,93)(79,89,99,84,94)(80,90,100,85,95), (1,42,60,77)(2,43,61,78)(3,44,62,79)(4,45,63,80)(5,46,64,81)(6,47,65,82)(7,48,66,83)(8,49,67,84)(9,50,68,85)(10,26,69,86)(11,27,70,87)(12,28,71,88)(13,29,72,89)(14,30,73,90)(15,31,74,91)(16,32,75,92)(17,33,51,93)(18,34,52,94)(19,35,53,95)(20,36,54,96)(21,37,55,97)(22,38,56,98)(23,39,57,99)(24,40,58,100)(25,41,59,76)>;

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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100), (1,6,11,16,21)(2,7,12,17,22)(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25)(26,41,31,46,36)(27,42,32,47,37)(28,43,33,48,38)(29,44,34,49,39)(30,45,35,50,40)(51,71,66,61,56)(52,72,67,62,57)(53,73,68,63,58)(54,74,69,64,59)(55,75,70,65,60)(76,86,96,81,91)(77,87,97,82,92)(78,88,98,83,93)(79,89,99,84,94)(80,90,100,85,95), (1,42,60,77)(2,43,61,78)(3,44,62,79)(4,45,63,80)(5,46,64,81)(6,47,65,82)(7,48,66,83)(8,49,67,84)(9,50,68,85)(10,26,69,86)(11,27,70,87)(12,28,71,88)(13,29,72,89)(14,30,73,90)(15,31,74,91)(16,32,75,92)(17,33,51,93)(18,34,52,94)(19,35,53,95)(20,36,54,96)(21,37,55,97)(22,38,56,98)(23,39,57,99)(24,40,58,100)(25,41,59,76) );

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),(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75),(76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)], [(1,6,11,16,21),(2,7,12,17,22),(3,8,13,18,23),(4,9,14,19,24),(5,10,15,20,25),(26,41,31,46,36),(27,42,32,47,37),(28,43,33,48,38),(29,44,34,49,39),(30,45,35,50,40),(51,71,66,61,56),(52,72,67,62,57),(53,73,68,63,58),(54,74,69,64,59),(55,75,70,65,60),(76,86,96,81,91),(77,87,97,82,92),(78,88,98,83,93),(79,89,99,84,94),(80,90,100,85,95)], [(1,42,60,77),(2,43,61,78),(3,44,62,79),(4,45,63,80),(5,46,64,81),(6,47,65,82),(7,48,66,83),(8,49,67,84),(9,50,68,85),(10,26,69,86),(11,27,70,87),(12,28,71,88),(13,29,72,89),(14,30,73,90),(15,31,74,91),(16,32,75,92),(17,33,51,93),(18,34,52,94),(19,35,53,95),(20,36,54,96),(21,37,55,97),(22,38,56,98),(23,39,57,99),(24,40,58,100),(25,41,59,76)]])

125 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E ··· 5I 10A 10B 10C 10D 20A ··· 20H 25A ··· 25T 25U ··· 25AN 50A ··· 50T 100A ··· 100AN order 1 2 4 4 5 5 5 5 5 ··· 5 10 10 10 10 20 ··· 20 25 ··· 25 25 ··· 25 50 ··· 50 100 ··· 100 size 1 5 5 5 1 1 1 1 4 ··· 4 5 5 5 5 5 ··· 5 1 ··· 1 4 ··· 4 5 ··· 5 5 ··· 5

125 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 4 type + + + image C1 C2 C4 C5 C10 C20 C25 C50 C100 F5 C5×F5 F5×C25 kernel F5×C25 D5×C25 C5×C25 C5×F5 C5×D5 C52 F5 D5 C5 C25 C5 C1 # reps 1 1 2 4 4 8 20 20 40 1 4 20

Matrix representation of F5×C25 in GL4(𝔽101) generated by

 81 0 0 0 0 81 0 0 0 0 81 0 0 0 0 81
,
 84 0 0 0 0 95 0 0 0 0 87 0 0 0 0 36
,
 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0
G:=sub<GL(4,GF(101))| [81,0,0,0,0,81,0,0,0,0,81,0,0,0,0,81],[84,0,0,0,0,95,0,0,0,0,87,0,0,0,0,36],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;

F5×C25 in GAP, Magma, Sage, TeX

F_5\times C_{25}
% in TeX

G:=Group("F5xC25");
// GroupNames label

G:=SmallGroup(500,15);
// by ID

G=gap.SmallGroup(500,15);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,106,5004,1014]);
// Polycyclic

G:=Group<a,b,c|a^25=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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