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G = C52⋊C20  order 500 = 22·53

The semidirect product of C52 and C20 acting faithfully

metabelian, supersoluble, monomial

Aliases: C52⋊C20, He5⋊1C4, C52⋊1F5, C5⋊D5.C10, C5⋊F5⋊C5, C5.2(C5×F5), C52⋊C10.1C2, SmallGroup(500,17)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — C52⋊C20
Chief series
C1 — C5 — C52 — C5⋊D5 — C52⋊C10 — C52⋊C20
Lower central
C52 — C52⋊C20
Upper central
C1

Generators and relations for C52⋊C20
 G = < a,b,c | a5=b5=c20=1, ab=ba, cac-1=a3b3, cbc-1=b3 >

C1
25C2
C5
5C5
5C5
20C5
25C4
5D5
25C10
25D5
C52
C52
4C52
5F5
25C20
25F5
C5⋊D5
5C5×D5
He5
C5⋊F5
5C5×F5
C52⋊C10
C52⋊C20

Character table of C52⋊C20

 class 124A4B5A5B5C5D5E5F5G5H5I5J10A10B10C10D20A20B20C20D20E20F20G20H
 size 1252525455552020202020252525252525252525252525
ρ111111111111111111111111111    trivial
ρ211-1-111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31-1i-i1111111111-1-1-1-1-ii-ii-ii-ii    linear of order 4
ρ41-1-ii1111111111-1-1-1-1i-ii-ii-ii-i    linear of order 4
ρ511-1-11ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ531ζ54ζ53ζ52ζ5-ζ52-ζ52-ζ54-ζ54-ζ5-ζ5-ζ53-ζ53    linear of order 10
ρ611111ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ51ζ53ζ5ζ54ζ52ζ54ζ54ζ53ζ53ζ52ζ52ζ5ζ5    linear of order 5
ρ711111ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ541ζ52ζ54ζ5ζ53ζ5ζ5ζ52ζ52ζ53ζ53ζ54ζ54    linear of order 5
ρ811111ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ521ζ5ζ52ζ53ζ54ζ53ζ53ζ5ζ5ζ54ζ54ζ52ζ52    linear of order 5
ρ911-1-11ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ541ζ52ζ54ζ5ζ53-ζ5-ζ5-ζ52-ζ52-ζ53-ζ53-ζ54-ζ54    linear of order 10
ρ1011-1-11ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ51ζ53ζ5ζ54ζ52-ζ54-ζ54-ζ53-ζ53-ζ52-ζ52-ζ5-ζ5    linear of order 10
ρ1111-1-11ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ521ζ5ζ52ζ53ζ54-ζ53-ζ53-ζ5-ζ5-ζ54-ζ54-ζ52-ζ52    linear of order 10
ρ1211111ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ531ζ54ζ53ζ52ζ5ζ52ζ52ζ54ζ54ζ5ζ5ζ53ζ53    linear of order 5
ρ131-1-ii1ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ521-ζ5-ζ52-ζ53-ζ54ζ4ζ53ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52ζ43ζ52    linear of order 20
ρ141-1i-i1ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ51-ζ53-ζ5-ζ54-ζ52ζ43ζ54ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5ζ4ζ5    linear of order 20
ρ151-1i-i1ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ521-ζ5-ζ52-ζ53-ζ54ζ43ζ53ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52ζ4ζ52    linear of order 20
ρ161-1-ii1ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ541-ζ52-ζ54-ζ5-ζ53ζ4ζ5ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54ζ43ζ54    linear of order 20
ρ171-1-ii1ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ51-ζ53-ζ5-ζ54-ζ52ζ4ζ54ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5ζ43ζ5    linear of order 20
ρ181-1i-i1ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ541-ζ52-ζ54-ζ5-ζ53ζ43ζ5ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54ζ4ζ54    linear of order 20
ρ191-1-ii1ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ531-ζ54-ζ53-ζ52-ζ5ζ4ζ52ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53ζ43ζ53    linear of order 20
ρ201-1i-i1ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ531-ζ54-ζ53-ζ52-ζ5ζ43ζ52ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53ζ4ζ53    linear of order 20
ρ21400044444-1-1-1-1-1000000000000    orthogonal lifted from F5
ρ22400044ζ544ζ54ζ524ζ53-ζ52-ζ5-ζ53-ζ54-1000000000000    complex lifted from C5×F5
ρ23400044ζ534ζ524ζ544ζ5-ζ54-ζ52-ζ5-ζ53-1000000000000    complex lifted from C5×F5
ρ24400044ζ54ζ544ζ534ζ52-ζ53-ζ54-ζ52-ζ5-1000000000000    complex lifted from C5×F5
ρ25400044ζ524ζ534ζ54ζ54-ζ5-ζ53-ζ54-ζ52-1000000000000    complex lifted from C5×F5
ρ2620000-5000000000000000000000    orthogonal faithful

Permutation representations of C52⋊C20
►On 25 points - transitive group 25T34
Generators in S25
(1 17 22 12 7)(2 9 18 16 15)(3 21 14 20 23)(4 25 6 8 19)(5 13 10 24 11)

(1 3 4 2 5)(6 18 10 22 14)(7 23 19 15 11)(8 16 24 12 20)(9 13 17 21 25)

(2 3 4 5)(6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25)
 
G:=sub<Sym(25)| (1,17,22,12,7)(2,9,18,16,15)(3,21,14,20,23)(4,25,6,8,19)(5,13,10,24,11), (1,3,4,2,5)(6,18,10,22,14)(7,23,19,15,11)(8,16,24,12,20)(9,13,17,21,25), (2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)>;
 
G:=Group( (1,17,22,12,7)(2,9,18,16,15)(3,21,14,20,23)(4,25,6,8,19)(5,13,10,24,11), (1,3,4,2,5)(6,18,10,22,14)(7,23,19,15,11)(8,16,24,12,20)(9,13,17,21,25), (2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25) );
 
G=PermutationGroup([[(1,17,22,12,7),(2,9,18,16,15),(3,21,14,20,23),(4,25,6,8,19),(5,13,10,24,11)], [(1,3,4,2,5),(6,18,10,22,14),(7,23,19,15,11),(8,16,24,12,20),(9,13,17,21,25)], [(2,3,4,5),(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)]])
 
G:=TransitiveGroup(25,34);
 

►On 25 points - transitive group 25T37
Generators in S25
(1 13 18 8 23)(2 24 9 19 14)(4 6 11 21 16)(5 7 12 22 17)

(1 18 23 13 8)(2 14 19 9 24)(3 10 15 25 20)(4 6 11 21 16)(5 22 7 17 12)

(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)
 
G:=sub<Sym(25)| (1,13,18,8,23)(2,24,9,19,14)(4,6,11,21,16)(5,7,12,22,17), (1,18,23,13,8)(2,14,19,9,24)(3,10,15,25,20)(4,6,11,21,16)(5,22,7,17,12), (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)>;
 
G:=Group( (1,13,18,8,23)(2,24,9,19,14)(4,6,11,21,16)(5,7,12,22,17), (1,18,23,13,8)(2,14,19,9,24)(3,10,15,25,20)(4,6,11,21,16)(5,22,7,17,12), (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) );
 
G=PermutationGroup([[(1,13,18,8,23),(2,24,9,19,14),(4,6,11,21,16),(5,7,12,22,17)], [(1,18,23,13,8),(2,14,19,9,24),(3,10,15,25,20),(4,6,11,21,16),(5,22,7,17,12)], [(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)]])
 
G:=TransitiveGroup(25,37);
 

Matrix representation of C52⋊C20 ►in GL20(ℤ)

10000000000000000000
01000000000000000000
00100000000000000000
00010000000000000000
00000100000000000000
00000010000000000000
00000001000000000000
0000-1-1-1-1000000000000
00000000001000000000
00000000000100000000
00000000-1-1-1-100000000
00000000100000000000
00000000000000010000
000000000000-1-1-1-10000
00000000000010000000
00000000000001000000
0000000000000000-1-1-1-1
00000000000000001000
00000000000000000100
00000000000000000010
,
01000000000000000000
00100000000000000000
00010000000000000000
-1-1-1-10000000000000000
00000100000000000000
00000010000000000000
00000001000000000000
0000-1-1-1-1000000000000
00000000010000000000
00000000001000000000
00000000000100000000
00000000-1-1-1-100000000
00000000000001000000
00000000000000100000
00000000000000010000
000000000000-1-1-1-10000
00000000000000000100
00000000000000000010
00000000000000000001
0000000000000000-1-1-1-1
,
00000000000000001000
00000000000000000001
00000000000000000100
0000000000000000-1-1-1-1
10000000000000000000
00010000000000000000
01000000000000000000
-1-1-1-10000000000000000
00001000000000000000
00000001000000000000
00000100000000000000
0000-1-1-1-1000000000000
00000000100000000000
00000000000100000000
00000000010000000000
00000000-1-1-1-100000000
00000000000010000000
00000000000000010000
00000000000001000000
000000000000-1-1-1-10000

G:=sub<GL(20,Integers())| [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0],[0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1],[0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] >;
 

C52⋊C20 in GAP, Magma, Sage, TeX 

C_5^2\rtimes C_{20}
 
% in TeX
 
G:=Group("C5^2:C20");
 
// GroupNames label
 
G:=SmallGroup(500,17);
 
// by ID
 
G=gap.SmallGroup(500,17);
 
# by ID
 
G:=PCGroup([5,-2,-5,-2,-5,-5,50,803,1208,173,5004,1014]);
 
// Polycyclic
 
G:=Group<a,b,c|a^5=b^5=c^20=1,a*b=b*a,c*a*c^-1=a^3*b^3,c*b*c^-1=b^3>;
 
// generators/relations
 

Export

Subgroup lattice of C52⋊C20 in TeX
Character table of C52⋊C20 in TeX

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