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## G = C2×C52⋊C10order 500 = 22·53

### Direct product of C2 and C52⋊C10

Aliases: C2×C52⋊C10, C522D10, He52C22, C5⋊D5⋊C10, (C5×C10)⋊C10, (C5×C10)⋊1D5, C52⋊(C2×C10), (C2×He5)⋊1C2, C10.5(C5×D5), C5.2(D5×C10), (C2×C5⋊D5)⋊C5, SmallGroup(500,30)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C52⋊C10
 Chief series C1 — C5 — C52 — He5 — C52⋊C10 — C2×C52⋊C10
 Lower central C52 — C2×C52⋊C10
 Upper central C1 — C2

Generators and relations for C2×C52⋊C10
G = < a,b,c,d | a2=b5=c5=d10=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c3, dcd-1=c-1 >

Smallest permutation representation of C2×C52⋊C10
On 50 points
Generators in S50
(1 2)(3 9)(4 10)(5 7)(6 8)(11 37)(12 38)(13 39)(14 40)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 41)(30 42)
(1 50 31 36 45)(2 28 15 20 23)(3 26 19 16 25)(4 30 11 14 21)(5 44 33 34 41)(6 46 39 38 49)(7 22 17 18 29)(8 24 13 12 27)(9 48 35 32 47)(10 42 37 40 43)
(1 10 5 6 9)(2 4 7 8 3)(11 17 13 19 15)(12 16 20 14 18)(21 29 27 25 23)(22 24 26 28 30)(31 37 33 39 35)(32 36 40 34 38)(41 49 47 45 43)(42 44 46 48 50)
(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)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 5A 5B 5C 5D 5E 5F 5G ··· 5P 10A 10B 10C 10D 10E 10F 10G ··· 10P 10Q ··· 10X order 1 2 2 2 5 5 5 5 5 5 5 ··· 5 10 10 10 10 10 10 10 ··· 10 10 ··· 10 size 1 1 25 25 2 2 5 5 5 5 10 ··· 10 2 2 5 5 5 5 10 ··· 10 25 ··· 25

44 irreducible representations

 dim 1 1 1 1 1 1 10 10 2 2 2 2 type + + + + + + + image C1 C2 C2 C5 C10 C10 C52⋊C10 C2×C52⋊C10 D5 D10 C5×D5 D5×C10 kernel C2×C52⋊C10 C52⋊C10 C2×He5 C2×C5⋊D5 C5⋊D5 C5×C10 C2 C1 C5×C10 C52 C10 C5 # reps 1 2 1 4 8 4 2 2 2 2 8 8

Matrix representation of C2×C52⋊C10 in GL12(𝔽11)

 10 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1
,
 3 1 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 10 7 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 10 7 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 10 7 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 10 7 0 0 0 1 0 0 0 0 0 0 0 0 0 0 10 7 0 0 0 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 0 0 1 0
,
 3 9 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 10 7 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 7 10 0 0 0 0 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 4 0 0 0 0 0 0 0 0 0 0 10 7 0 0 0 0 0 0

G:=sub<GL(12,GF(11))| [10,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1],[3,10,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,1,7,0,0,0,10,0,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,1,7,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,0,10,0],[3,0,0,0,0,0,0,0,0,0,0,0,9,8,0,0,0,0,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,10,0,0,0,0,0,0,0,0,0,0,4,7,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,4,0,0,0,0,0,0,0,0,0,0,10,4,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0] >;

C2×C52⋊C10 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes C_{10}
% in TeX

G:=Group("C2xC5^2:C10");
// GroupNames label

G:=SmallGroup(500,30);
// by ID

G=gap.SmallGroup(500,30);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,1603,613,10004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^5=d^10=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c^3,d*c*d^-1=c^-1>;
// generators/relations

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