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## G = D5×C50order 500 = 22·53

### Direct product of C50 and D5

Aliases: D5×C50, C10⋊C50, C5⋊(C2×C50), (C5×C50)⋊1C2, (D5×C10).C5, (C5×C25)⋊2C22, C10.9(C5×D5), C5.4(D5×C10), (C5×C10).6C10, (C5×D5).4C10, C52.2(C2×C10), SmallGroup(500,29)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C50
 Chief series C1 — C5 — C52 — C5×C25 — D5×C25 — D5×C50
 Lower central C5 — D5×C50
 Upper central C1 — C50

Generators and relations for D5×C50
G = < a,b,c | a50=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C50
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 21 41 11 31)(2 22 42 12 32)(3 23 43 13 33)(4 24 44 14 34)(5 25 45 15 35)(6 26 46 16 36)(7 27 47 17 37)(8 28 48 18 38)(9 29 49 19 39)(10 30 50 20 40)(51 81 61 91 71)(52 82 62 92 72)(53 83 63 93 73)(54 84 64 94 74)(55 85 65 95 75)(56 86 66 96 76)(57 87 67 97 77)(58 88 68 98 78)(59 89 69 99 79)(60 90 70 100 80)
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 93)(50 94)

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

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

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

200 conjugacy classes

 class 1 2A 2B 2C 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10N 10O ··· 10V 25A ··· 25T 25U ··· 25BH 50A ··· 50T 50U ··· 50BH 50BI ··· 50CV order 1 2 2 2 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 25 ··· 25 25 ··· 25 50 ··· 50 50 ··· 50 50 ··· 50 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 C2 C5 C10 C10 C25 C50 C50 D5 D10 C5×D5 D5×C10 D5×C25 D5×C50 kernel D5×C50 D5×C25 C5×C50 D5×C10 C5×D5 C5×C10 D10 D5 C10 C50 C25 C10 C5 C2 C1 # reps 1 2 1 4 8 4 20 40 20 2 2 8 8 40 40

Matrix representation of D5×C50 in GL2(𝔽101) generated by

 45 0 0 45
,
 36 0 0 87
,
 0 14 65 0
G:=sub<GL(2,GF(101))| [45,0,0,45],[36,0,0,87],[0,65,14,0] >;

D5×C50 in GAP, Magma, Sage, TeX

D_5\times C_{50}
% in TeX

G:=Group("D5xC50");
// GroupNames label

G:=SmallGroup(500,29);
// by ID

G=gap.SmallGroup(500,29);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,87,10004]);
// Polycyclic

G:=Group<a,b,c|a^50=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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