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## G = D5×C5⋊D5order 500 = 22·53

### Direct product of D5 and C5⋊D5

Aliases: D5×C5⋊D5, C524D10, C532C22, C51D52, (C5×D5)⋊D5, C53⋊C2⋊C2, (D5×C52)⋊2C2, C51(C2×C5⋊D5), (C5×C5⋊D5)⋊2C2, SmallGroup(500,51)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×C5⋊D5
G = < a,b,c,d,e | a5=b2=c5=d5=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 1616 in 128 conjugacy classes, 28 normal (10 characteristic)
C1, C2, C22, C5, C5, C5, D5, D5, C10, D10, C52, C52, C52, C5×D5, C5×D5, C5⋊D5, C5⋊D5, C5×C10, D52, C2×C5⋊D5, C53, D5×C52, C5×C5⋊D5, C53⋊C2, D5×C5⋊D5
Quotients: C1, C2, C22, D5, D10, C5⋊D5, D52, C2×C5⋊D5, D5×C5⋊D5

Smallest permutation representation of D5×C5⋊D5
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 28)(2 27)(3 26)(4 30)(5 29)(6 46)(7 50)(8 49)(9 48)(10 47)(11 41)(12 45)(13 44)(14 43)(15 42)(16 36)(17 40)(18 39)(19 38)(20 37)(21 31)(22 35)(23 34)(24 33)(25 32)
(1 24 19 14 9)(2 25 20 15 10)(3 21 16 11 6)(4 22 17 12 7)(5 23 18 13 8)(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 23 17 11 10)(2 24 18 12 6)(3 25 19 13 7)(4 21 20 14 8)(5 22 16 15 9)(26 32 38 44 50)(27 33 39 45 46)(28 34 40 41 47)(29 35 36 42 48)(30 31 37 43 49)
(1 48)(2 49)(3 50)(4 46)(5 47)(6 30)(7 26)(8 27)(9 28)(10 29)(11 35)(12 31)(13 32)(14 33)(15 34)(16 40)(17 36)(18 37)(19 38)(20 39)(21 45)(22 41)(23 42)(24 43)(25 44)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 5A ··· 5N 5O ··· 5AL 10A ··· 10L 10M 10N order 1 2 2 2 5 ··· 5 5 ··· 5 10 ··· 10 10 10 size 1 5 25 125 2 ··· 2 4 ··· 4 10 ··· 10 50 50

56 irreducible representations

 dim 1 1 1 1 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 D5 D5 D10 D52 kernel D5×C5⋊D5 D5×C52 C5×C5⋊D5 C53⋊C2 C5×D5 C5⋊D5 C52 C5 # reps 1 1 1 1 12 2 14 24

Matrix representation of D5×C5⋊D5 in GL6(𝔽11)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 10 0 0 0 0 1 0
,
 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 0 0 3 10 0 0 0 0 8 8
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 10 0 0 0 0 1 3 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 1 0 0 0 0 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 1 0 0 0 0 3 8 0 0 0 0 0 0 3 1 0 0 0 0 3 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(11))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,1,0,0,0,0,10,0],[10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,3,8,0,0,0,0,10,8],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,10,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,10,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,1,8,0,0,0,0,0,0,3,3,0,0,0,0,1,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D5×C5⋊D5 in GAP, Magma, Sage, TeX

D_5\times C_5\rtimes D_5
% in TeX

G:=Group("D5xC5:D5");
// GroupNames label

G:=SmallGroup(500,51);
// by ID

G=gap.SmallGroup(500,51);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,127,808,10004]);
// Polycyclic

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

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