Copied to
clipboard

G = C52⋊5D10  order 500 = 22·53

2nd semidirect product of C52 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, A-group

Aliases: C52⋊5D10, C53⋊3C22, C5⋊2D52, C5⋊D5⋊2D5, (C5×C5⋊D5)⋊3C2, SmallGroup(500,52)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C53 — C52⋊5D10
Chief series
C1 — C5 — C52 — C53 — C5×C5⋊D5 — C52⋊5D10
Lower central
C53 — C52⋊5D10
Upper central
C1

Generators and relations for C52⋊5D10
 G = < a,b,c,d | a5=b5=c10=d2=1, ab=ba, cac-1=a-1, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 616 in 72 conjugacy classes, 15 normal (3 characteristic)
C1, C2, C22, C5, C5, D5, C10, D10, C52, C52, C5×D5, C5⋊D5, D52, C53, C5×C5⋊D5, C52⋊5D10
Quotients: C1, C2, C22, D5, D10, D52, C52⋊5D10

Permutation representations of C52⋊5D10
►On 20 points - transitive group 20T128
Generators in S20
(1 7 3 9 5)(2 6 10 4 8)(11 17 13 19 15)(12 16 20 14 18)

(1 9 7 5 3)(2 4 6 8 10)(11 13 15 17 19)(12 20 18 16 14)

(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)

(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 20)(8 19)(9 18)(10 17)
 
G:=sub<Sym(20)| (1,7,3,9,5)(2,6,10,4,8)(11,17,13,19,15)(12,16,20,14,18), (1,9,7,5,3)(2,4,6,8,10)(11,13,15,17,19)(12,20,18,16,14), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,20)(8,19)(9,18)(10,17)>;
 
G:=Group( (1,7,3,9,5)(2,6,10,4,8)(11,17,13,19,15)(12,16,20,14,18), (1,9,7,5,3)(2,4,6,8,10)(11,13,15,17,19)(12,20,18,16,14), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,20)(8,19)(9,18)(10,17) );
 
G=PermutationGroup([[(1,7,3,9,5),(2,6,10,4,8),(11,17,13,19,15),(12,16,20,14,18)], [(1,9,7,5,3),(2,4,6,8,10),(11,13,15,17,19),(12,20,18,16,14)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,20),(8,19),(9,18),(10,17)]])
 
G:=TransitiveGroup(20,128);
 

44 conjugacy classes

class 1 2A2B2C5A···5F5G···5AH10A···10F
order12225···55···510···10
size12525252···24···450···50

44 irreducible representations

dim112244
type+++++
imageC1C2D5D10D52C52⋊5D10
kernelC52⋊5D10C5×C5⋊D5C5⋊D5C52C5C1
# reps13661216

Matrix representation of C52⋊5D10 ►in GL6(𝔽11)

100000
010000
0071000
001000
000010
000001
,
100000
010000
001000
000100
0000310
000010
,
740000
710000
001000
0071000
000010
0000310
,
710000
740000
001000
000100
000010
0000310

G:=sub<GL(6,GF(11))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,1,0,0,0,0,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[7,7,0,0,0,0,4,1,0,0,0,0,0,0,1,7,0,0,0,0,0,10,0,0,0,0,0,0,1,3,0,0,0,0,0,10],[7,7,0,0,0,0,1,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,10] >;
 

C52⋊5D10 in GAP, Magma, Sage, TeX 

C_5^2\rtimes_5D_{10}
 
% in TeX
 
G:=Group("C5^2:5D10");
 
// GroupNames label
 
G:=SmallGroup(500,52);
 
// by ID
 
G=gap.SmallGroup(500,52);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,242,127,808,10004]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^5=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁