Copied to
clipboard

G = Dic5×C25order 500 = 22·53

Direct product of C25 and Dic5

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

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

C1C5 — Dic5×C25
C1C5C52C5×C10C5×C50 — Dic5×C25
C5 — Dic5×C25
C1C50

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

2C5
2C5
5C4
2C10
2C10
2C25
2C25
5C20
2C50
2C50
5C100

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 4A4B5A5B5C5D5E···5N10A10B10C10D10E···10N20A···20H25A···25T25U···25BH50A···50T50U···50BH100A···100AN
order124455555···51010101010···1020···2025···2525···2550···5050···50100···100
size115511112···211112···25···51···12···21···12···25···5

200 irreducible representations

dim111111111222222
type+++-
imageC1C2C4C5C10C20C25C50C100D5Dic5C5×D5C5×Dic5D5×C25Dic5×C25
kernelDic5×C25C5×C50C5×C25C5×Dic5C5×C10C52Dic5C10C5C50C25C10C5C2C1
# reps11244820204022884040

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

540
054
,
140
065
,
01
1000
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

Export

Subgroup lattice of Dic5×C25 in TeX

׿
×
𝔽