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G = Dic5xC25order 500 = 22·53

Direct product of C25 and Dic5

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

Aliases: Dic5xC25, C10.C50, C5:2C100, C50.4D5, C52.3C20, (C5xC25):7C4, C2.(D5xC25), (C5xC50).1C2, C10.8(C5xD5), (C5xDic5).C5, (C5xC10).5C10, C5.4(C5xDic5), SmallGroup(500,7)

Series: Derived Chief Lower central Upper central

C1C5 — Dic5xC25
C1C5C52C5xC10C5xC50 — Dic5xC25
C5 — Dic5xC25
C1C50

Generators and relations for Dic5xC25
 G = < a,b,c | a25=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 46 in 26 conjugacy classes, 15 normal (all characteristic)
Quotients: C1, C2, C4, C5, D5, C10, Dic5, C20, C25, C50, C5xD5, C100, C5xDic5, D5xC25, Dic5xC25
2C5
2C5
5C4
2C10
2C10
2C25
2C25
5C20
2C50
2C50
5C100

Smallest permutation representation of Dic5xC25
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 53 16 68 6 58 21 73 11 63)(2 54 17 69 7 59 22 74 12 64)(3 55 18 70 8 60 23 75 13 65)(4 56 19 71 9 61 24 51 14 66)(5 57 20 72 10 62 25 52 15 67)(26 100 36 85 46 95 31 80 41 90)(27 76 37 86 47 96 32 81 42 91)(28 77 38 87 48 97 33 82 43 92)(29 78 39 88 49 98 34 83 44 93)(30 79 40 89 50 99 35 84 45 94)
(1 85 58 41)(2 86 59 42)(3 87 60 43)(4 88 61 44)(5 89 62 45)(6 90 63 46)(7 91 64 47)(8 92 65 48)(9 93 66 49)(10 94 67 50)(11 95 68 26)(12 96 69 27)(13 97 70 28)(14 98 71 29)(15 99 72 30)(16 100 73 31)(17 76 74 32)(18 77 75 33)(19 78 51 34)(20 79 52 35)(21 80 53 36)(22 81 54 37)(23 82 55 38)(24 83 56 39)(25 84 57 40)

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

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

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

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+++-
imageC1C2C4C5C10C20C25C50C100D5Dic5C5xD5C5xDic5D5xC25Dic5xC25
kernelDic5xC25C5xC50C5xC25C5xDic5C5xC10C52Dic5C10C5C50C25C10C5C2C1
# reps11244820204022884040

Matrix representation of Dic5xC25 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] >;

Dic5xC25 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 Dic5xC25 in TeX

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