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## G = C5×D25order 250 = 2·53

### Direct product of C5 and D25

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C5×D25
 Chief series C1 — C5 — C25 — C5×C25 — C5×D25
 Lower central C25 — C5×D25
 Upper central C1 — C5

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

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

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

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

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

C5×D25 is a maximal subgroup of   D25.D5

70 conjugacy classes

 class 1 2 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 25A ··· 25AX order 1 2 5 5 5 5 5 ··· 5 10 10 10 10 25 ··· 25 size 1 25 1 1 1 1 2 ··· 2 25 25 25 25 2 ··· 2

70 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + image C1 C2 C5 C10 D5 D25 C5×D5 C5×D25 kernel C5×D25 C5×C25 D25 C25 C52 C5 C5 C1 # reps 1 1 4 4 2 10 8 40

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

 84 0 0 84
,
 78 0 34 79
,
 85 53 100 16
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

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