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G = C2×C52⋊C10  order 500 = 22·53

Direct product of C2 and C52⋊C10

direct product, metabelian, supersoluble, monomial

Aliases: C2×C52⋊C10, C52⋊2D10, He5⋊2C22, 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 >

C1
C2
25C2
25C2
C5
5C5
5C5
10C5
10C5
25C22
C10
5D5
5C10
5C10
5D5
10C10
10C10
25C10
25D5
25D5
25C10
C52
C52
2C52
2C52
5D10
25D10
25C2×C10
C5×C10
C5⋊D5
C5⋊D5
C5×C10
2C5×C10
2C5×C10
5C5×D5
5C5×D5
He5
C2×C5⋊D5
5D5×C10
C52⋊C10
C52⋊C10
C2×He5
C2×C52⋊C10

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 2A2B2C5A5B5C5D5E5F5G···5P10A10B10C10D10E10F10G···10P10Q···10X
order12225555555···510101010101010···1010···10
size11252522555510···1022555510···1025···25

44 irreducible representations

dim11111110102222
type+++++++
imageC1C2C2C5C10C10C52⋊C10C2×C52⋊C10D5D10C5×D5D5×C10
kernelC2×C52⋊C10C52⋊C10C2×He5C2×C5⋊D5C5⋊D5C5×C10C2C1C5×C10C52C10C5
# reps121484222288

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

1000000000000
0100000000000
001000000000
000100000000
000010000000
000001000000
000000100000
000000010000
000000001000
000000000100
000000000010
000000000001
,
310000000000
1000000000000
000001000000
0000107000000
000000010000
0000001070000
000000000100
0000000010700
000000000001
0000000000107
000100000000
0010700000000
,
100000000000
010000000000
0071000000000
001000000000
0000710000000
000010000000
0000007100000
000000100000
0000000071000
000000001000
0000000000710
000000000010
,
390000000000
080000000000
001000000000
0071000000000
0000000000107
000000000001
0000000071000
000000004400
000000010000
000000100000
000044000000
0000107000000

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
 

Export

Subgroup lattice of C2×C52⋊C10 in TeX

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