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G = D5×C50order 500 = 22·53

Direct product of C50 and D5

direct product, metacyclic, supersoluble, monomial, A-group

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

C1C5 — D5×C50
C1C5C52C5×C25D5×C25 — D5×C50
C5 — D5×C50
C1C50

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

5C2
5C2
2C5
2C5
5C22
2C10
2C10
5C10
5C10
2C25
2C25
5C2×C10
2C50
2C50
5C50
5C50
5C2×C50

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 2A2B2C5A5B5C5D5E···5N10A10B10C10D10E···10N10O···10V25A···25T25U···25BH50A···50T50U···50BH50BI···50CV
order122255555···51010101010···1010···1025···2525···2550···5050···5050···50
size115511112···211112···25···51···12···21···12···25···5

200 irreducible representations

dim111111111222222
type+++++
imageC1C2C2C5C10C10C25C50C50D5D10C5×D5D5×C10D5×C25D5×C50
kernelD5×C50D5×C25C5×C50D5×C10C5×D5C5×C10D10D5C10C50C25C10C5C2C1
# reps12148420402022884040

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

450
045
,
360
087
,
014
650
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

Export

Subgroup lattice of D5×C50 in TeX

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