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G = C10×D25order 500 = 22·53

Direct product of C10 and D25

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

Aliases: C10×D25, C503C10, C52.3D10, (C5×C50)⋊2C2, C253(C2×C10), (C5×C25)⋊3C22, (C5×C10).7D5, C5.1(D5×C10), C10.4(C5×D5), SmallGroup(500,28)

Series: Derived Chief Lower central Upper central

C1C25 — C10×D25
C1C5C25C5×C25C5×D25 — C10×D25
C25 — C10×D25
C1C10

Generators and relations for C10×D25
 G = < a,b,c | a10=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >

25C2
25C2
2C5
2C5
25C22
2C10
2C10
5D5
5D5
25C10
25C10
2C25
2C25
5D10
25C2×C10
2C50
2C50
5C5×D5
5C5×D5
5D5×C10

Smallest permutation representation of C10×D25
On 100 points
Generators in S100
(1 28 11 38 21 48 6 33 16 43)(2 29 12 39 22 49 7 34 17 44)(3 30 13 40 23 50 8 35 18 45)(4 31 14 41 24 26 9 36 19 46)(5 32 15 42 25 27 10 37 20 47)(51 79 66 94 56 84 71 99 61 89)(52 80 67 95 57 85 72 100 62 90)(53 81 68 96 58 86 73 76 63 91)(54 82 69 97 59 87 74 77 64 92)(55 83 70 98 60 88 75 78 65 93)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 63)(7 62)(8 61)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 75)(20 74)(21 73)(22 72)(23 71)(24 70)(25 69)(26 98)(27 97)(28 96)(29 95)(30 94)(31 93)(32 92)(33 91)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(41 83)(42 82)(43 81)(44 80)(45 79)(46 78)(47 77)(48 76)(49 100)(50 99)

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

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

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

140 conjugacy classes

class 1 2A2B2C5A5B5C5D5E···5N10A10B10C10D10E···10N10O···10V25A···25AX50A···50AX
order122255555···51010101010···1010···1025···2550···50
size11252511112···211112···225···252···22···2

140 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C5C10C10D5D10D25C5×D5D50D5×C10C5×D25C10×D25
kernelC10×D25C5×D25C5×C50D50D25C50C5×C10C52C10C10C5C5C2C1
# reps121484221081084040

Matrix representation of C10×D25 in GL3(𝔽101) generated by

6500
0840
0084
,
100
08820
0031
,
100
08820
03213
G:=sub<GL(3,GF(101))| [65,0,0,0,84,0,0,0,84],[1,0,0,0,88,0,0,20,31],[1,0,0,0,88,32,0,20,13] >;

C10×D25 in GAP, Magma, Sage, TeX

C_{10}\times D_{25}
% in TeX

G:=Group("C10xD25");
// GroupNames label

G:=SmallGroup(500,28);
// by ID

G=gap.SmallGroup(500,28);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,3603,418,10004]);
// Polycyclic

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

Export

Subgroup lattice of C10×D25 in TeX

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