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## G = D5×D25order 500 = 22·53

### Direct product of D5 and D25

Aliases: D5×D25, C51D50, C251D10, C52.2D10, C25⋊D5⋊C2, C5.1D52, (C5×D25)⋊C2, (D5×C25)⋊C2, (C5×C25)⋊C22, (C5×D5).3D5, SmallGroup(500,26)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C25 — D5×D25
 Chief series C1 — C5 — C52 — C5×C25 — D5×C25 — D5×D25
 Lower central C5×C25 — D5×D25
 Upper central C1

Generators and relations for D5×D25
G = < a,b,c,d | a5=b2=c25=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

5C2
25C2
125C2
2C5
2C5
125C22
5C10
5D5
25D5
25C10
25D5
50D5
50D5
2C25
2C25
25D10
25D10
5C50
5D25
10D25
10D25
5D50
5D52

Smallest permutation representation of D5×D25
On 50 points
Generators in S50
(1 11 21 6 16)(2 12 22 7 17)(3 13 23 8 18)(4 14 24 9 19)(5 15 25 10 20)(26 41 31 46 36)(27 42 32 47 37)(28 43 33 48 38)(29 44 34 49 39)(30 45 35 50 40)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 26)(21 27)(22 28)(23 29)(24 30)(25 31)
(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 46)(2 45)(3 44)(4 43)(5 42)(6 41)(7 40)(8 39)(9 38)(10 37)(11 36)(12 35)(13 34)(14 33)(15 32)(16 31)(17 30)(18 29)(19 28)(20 27)(21 26)(22 50)(23 49)(24 48)(25 47)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 25A ··· 25J 25K ··· 25AD 50A ··· 50J order 1 2 2 2 5 5 5 5 5 5 5 5 10 10 10 10 25 ··· 25 25 ··· 25 50 ··· 50 size 1 5 25 125 2 2 2 2 4 4 4 4 10 10 50 50 2 ··· 2 4 ··· 4 10 ··· 10

56 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 D5 D5 D10 D10 D25 D50 D52 D5×D25 kernel D5×D25 C5×D25 D5×C25 C25⋊D5 D25 C5×D5 C25 C52 D5 C5 C5 C1 # reps 1 1 1 1 2 2 2 2 10 10 4 20

Matrix representation of D5×D25 in GL4(𝔽101) generated by

 78 100 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 23 1 0 0 78 78 0 0 0 0 100 0 0 0 0 100
,
 1 0 0 0 0 1 0 0 0 0 50 25 0 0 16 6
,
 100 0 0 0 0 100 0 0 0 0 51 76 0 0 3 50
G:=sub<GL(4,GF(101))| [78,1,0,0,100,0,0,0,0,0,1,0,0,0,0,1],[23,78,0,0,1,78,0,0,0,0,100,0,0,0,0,100],[1,0,0,0,0,1,0,0,0,0,50,16,0,0,25,6],[100,0,0,0,0,100,0,0,0,0,51,3,0,0,76,50] >;

D5×D25 in GAP, Magma, Sage, TeX

D_5\times D_{25}
% in TeX

G:=Group("D5xD25");
// GroupNames label

G:=SmallGroup(500,26);
// by ID

G=gap.SmallGroup(500,26);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,877,1512,1603,5009]);
// Polycyclic

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

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