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## G = D5×C52order 250 = 2·53

### Direct product of C52 and D5

Aliases: D5×C52, C531C2, C523C10, C5⋊(C5×C10), SmallGroup(250,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C52
 Chief series C1 — C5 — C52 — C53 — D5×C52
 Lower central C5 — D5×C52
 Upper central C1 — C52

Generators and relations for D5×C52
G = < a,b,c,d | a5=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 112 in 56 conjugacy classes, 24 normal (6 characteristic)
C1, C2, C5, C5, C5, D5, C10, C52, C52, C52, C5×D5, C5×C10, C53, D5×C52
Quotients: C1, C2, C5, D5, C10, C52, C5×D5, C5×C10, D5×C52

Smallest permutation representation of D5×C52
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 35 40 45 26)(2 31 36 41 27)(3 32 37 42 28)(4 33 38 43 29)(5 34 39 44 30)(6 11 16 21 50)(7 12 17 22 46)(8 13 18 23 47)(9 14 19 24 48)(10 15 20 25 49)
(1 26 45 40 35)(2 27 41 36 31)(3 28 42 37 32)(4 29 43 38 33)(5 30 44 39 34)(6 11 16 21 50)(7 12 17 22 46)(8 13 18 23 47)(9 14 19 24 48)(10 15 20 25 49)
(1 50)(2 46)(3 47)(4 48)(5 49)(6 35)(7 31)(8 32)(9 33)(10 34)(11 40)(12 36)(13 37)(14 38)(15 39)(16 45)(17 41)(18 42)(19 43)(20 44)(21 26)(22 27)(23 28)(24 29)(25 30)

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

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

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

D5×C52 is a maximal subgroup of   C53⋊C4

100 conjugacy classes

 class 1 2 5A ··· 5X 5Y ··· 5BV 10A ··· 10X order 1 2 5 ··· 5 5 ··· 5 10 ··· 10 size 1 5 1 ··· 1 2 ··· 2 5 ··· 5

100 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C5 C10 D5 C5×D5 kernel D5×C52 C53 C5×D5 C52 C52 C5 # reps 1 1 24 24 2 48

Matrix representation of D5×C52 in GL3(𝔽11) generated by

 1 0 0 0 9 0 0 0 9
,
 9 0 0 0 4 0 0 0 4
,
 1 0 0 0 3 0 0 0 4
,
 10 0 0 0 0 4 0 3 0
G:=sub<GL(3,GF(11))| [1,0,0,0,9,0,0,0,9],[9,0,0,0,4,0,0,0,4],[1,0,0,0,3,0,0,0,4],[10,0,0,0,0,3,0,4,0] >;

D5×C52 in GAP, Magma, Sage, TeX

D_5\times C_5^2
% in TeX

G:=Group("D5xC5^2");
// GroupNames label

G:=SmallGroup(250,12);
// by ID

G=gap.SmallGroup(250,12);
# by ID

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

G:=Group<a,b,c,d|a^5=b^5=c^5=d^2=1,a*b=b*a,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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