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G = C5×D25order 250 = 2·53

Direct product of C5 and D25

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

Aliases: C5×D25, C253C10, C52.2D5, (C5×C25)⋊2C2, C5.1(C5×D5), SmallGroup(250,3)

Series: Derived Chief Lower central Upper central

C1C25 — C5×D25
C1C5C25C5×C25 — C5×D25
C25 — C5×D25
C1C5

Generators and relations for C5×D25
 G = < a,b,c | a5=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >

25C2
2C5
2C5
5D5
25C10
2C25
2C25
5C5×D5

Smallest permutation representation of C5×D25
On 50 points
Generators in S50
(1 6 11 16 21)(2 7 12 17 22)(3 8 13 18 23)(4 9 14 19 24)(5 10 15 20 25)(26 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)
(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 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 40)

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

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

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

C5×D25 is a maximal subgroup of   D25.D5

70 conjugacy classes

class 1  2 5A5B5C5D5E···5N10A10B10C10D25A···25AX
order1255555···51010101025···25
size12511112···2252525252···2

70 irreducible representations

dim11112222
type++++
imageC1C2C5C10D5D25C5×D5C5×D25
kernelC5×D25C5×C25D25C25C52C5C5C1
# reps1144210840

Matrix representation of C5×D25 in GL2(𝔽101) generated by

840
084
,
780
3479
,
8553
10016
G:=sub<GL(2,GF(101))| [84,0,0,84],[78,34,0,79],[85,100,53,16] >;

C5×D25 in GAP, Magma, Sage, TeX

C_5\times D_{25}
% in TeX

G:=Group("C5xD25");
// GroupNames label

G:=SmallGroup(250,3);
// by ID

G=gap.SmallGroup(250,3);
# by ID

G:=PCGroup([4,-2,-5,-5,-5,1082,250,3203]);
// Polycyclic

G:=Group<a,b,c|a^5=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D25 in TeX

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