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## G = Dic5×C25order 500 = 22·53

### Direct product of C25 and Dic5

Aliases: Dic5×C25, C10.C50, C52C100, C50.4D5, C52.3C20, (C5×C25)⋊7C4, C2.(D5×C25), (C5×C50).1C2, C10.8(C5×D5), (C5×Dic5).C5, (C5×C10).5C10, C5.4(C5×Dic5), SmallGroup(500,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — Dic5×C25
 Chief series C1 — C5 — C52 — C5×C10 — C5×C50 — Dic5×C25
 Lower central C5 — Dic5×C25
 Upper central C1 — C50

Generators and relations for Dic5×C25
G = < a,b,c | a25=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Dic5×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 40 16 30 6 45 21 35 11 50)(2 41 17 31 7 46 22 36 12 26)(3 42 18 32 8 47 23 37 13 27)(4 43 19 33 9 48 24 38 14 28)(5 44 20 34 10 49 25 39 15 29)(51 99 61 84 71 94 56 79 66 89)(52 100 62 85 72 95 57 80 67 90)(53 76 63 86 73 96 58 81 68 91)(54 77 64 87 74 97 59 82 69 92)(55 78 65 88 75 98 60 83 70 93)
(1 63 45 81)(2 64 46 82)(3 65 47 83)(4 66 48 84)(5 67 49 85)(6 68 50 86)(7 69 26 87)(8 70 27 88)(9 71 28 89)(10 72 29 90)(11 73 30 91)(12 74 31 92)(13 75 32 93)(14 51 33 94)(15 52 34 95)(16 53 35 96)(17 54 36 97)(18 55 37 98)(19 56 38 99)(20 57 39 100)(21 58 40 76)(22 59 41 77)(23 60 42 78)(24 61 43 79)(25 62 44 80)

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

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

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

200 conjugacy classes

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

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + - image C1 C2 C4 C5 C10 C20 C25 C50 C100 D5 Dic5 C5×D5 C5×Dic5 D5×C25 Dic5×C25 kernel Dic5×C25 C5×C50 C5×C25 C5×Dic5 C5×C10 C52 Dic5 C10 C5 C50 C25 C10 C5 C2 C1 # reps 1 1 2 4 4 8 20 20 40 2 2 8 8 40 40

Matrix representation of Dic5×C25 in GL2(𝔽101) generated by

 54 0 0 54
,
 14 0 0 65
,
 0 1 100 0
G:=sub<GL(2,GF(101))| [54,0,0,54],[14,0,0,65],[0,100,1,0] >;

Dic5×C25 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_{25}
% in TeX

G:=Group("Dic5xC25");
// GroupNames label

G:=SmallGroup(500,7);
// by ID

G=gap.SmallGroup(500,7);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,106,10004]);
// Polycyclic

G:=Group<a,b,c|a^25=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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