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G = D5×C25order 250 = 2·53

Direct product of C25 and D5

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

Aliases: D5×C25, C5⋊C50, C52.2C10, (C5×C25)⋊1C2, (C5×D5).C5, C5.4(C5×D5), SmallGroup(250,4)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C25
C1C5C52C5×C25 — D5×C25
C5 — D5×C25
C1C25

Generators and relations for D5×C25
 G = < a,b,c | a25=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
2C5
2C5
5C10
2C25
2C25
5C50

Smallest permutation representation of D5×C25
On 50 points
Generators in S50
(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)
(1 21 16 11 6)(2 22 17 12 7)(3 23 18 13 8)(4 24 19 14 9)(5 25 20 15 10)(26 31 36 41 46)(27 32 37 42 47)(28 33 38 43 48)(29 34 39 44 49)(30 35 40 45 50)
(1 46)(2 47)(3 48)(4 49)(5 50)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)

G:=sub<Sym(50)| (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), (1,21,16,11,6)(2,22,17,12,7)(3,23,18,13,8)(4,24,19,14,9)(5,25,20,15,10)(26,31,36,41,46)(27,32,37,42,47)(28,33,38,43,48)(29,34,39,44,49)(30,35,40,45,50), (1,46)(2,47)(3,48)(4,49)(5,50)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)>;

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), (1,21,16,11,6)(2,22,17,12,7)(3,23,18,13,8)(4,24,19,14,9)(5,25,20,15,10)(26,31,36,41,46)(27,32,37,42,47)(28,33,38,43,48)(29,34,39,44,49)(30,35,40,45,50), (1,46)(2,47)(3,48)(4,49)(5,50)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45) );

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)], [(1,21,16,11,6),(2,22,17,12,7),(3,23,18,13,8),(4,24,19,14,9),(5,25,20,15,10),(26,31,36,41,46),(27,32,37,42,47),(28,33,38,43,48),(29,34,39,44,49),(30,35,40,45,50)], [(1,46),(2,47),(3,48),(4,49),(5,50),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45)])

D5×C25 is a maximal subgroup of   D5.D25

100 conjugacy classes

class 1  2 5A5B5C5D5E···5N10A10B10C10D25A···25T25U···25BH50A···50T
order1255555···51010101025···2525···2550···50
size1511112···255551···12···25···5

100 irreducible representations

dim111111222
type+++
imageC1C2C5C10C25C50D5C5×D5D5×C25
kernelD5×C25C5×C25C5×D5C52D5C5C25C5C1
# reps114420202840

Matrix representation of D5×C25 in GL2(𝔽101) generated by

790
079
,
840
9795
,
257
8476
G:=sub<GL(2,GF(101))| [79,0,0,79],[84,97,0,95],[25,84,7,76] >;

D5×C25 in GAP, Magma, Sage, TeX

D_5\times C_{25}
% in TeX

G:=Group("D5xC25");
// GroupNames label

G:=SmallGroup(250,4);
// by ID

G=gap.SmallGroup(250,4);
# by ID

G:=PCGroup([4,-2,-5,-5,-5,45,3203]);
// Polycyclic

G:=Group<a,b,c|a^25=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×C25 in TeX

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