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G = C25⋊C20order 500 = 22·53

The semidirect product of C25 and C20 acting faithfully

metacyclic, supersoluble, monomial

Aliases: C25⋊C20, C52.F5, D25.C10, 5- 1+2⋊C4, C25⋊C4⋊C5, C25⋊C10.C2, C5.3(C5×F5), Aut(D25), Hol(C25), SmallGroup(500,18)

Series: Derived Chief Lower central Upper central

C1C25 — C25⋊C20
C1C5C25D25C25⋊C10 — C25⋊C20
C25 — C25⋊C20
C1

Generators and relations for C25⋊C20
 G = < a,b | a25=b20=1, bab-1=a3 >

25C2
5C5
25C4
5D5
25C10
4C25
5F5
25C20
5C5×D5
5C5×F5

Character table of C25⋊C20

 class 124A4B5A5B5C5D5E10A10B10C10D20A20B20C20D20E20F20G20H25A25B25C25D25E
 size 1252525455552525252525252525252525252020202020
ρ111111111111111111111111111    trivial
ρ211-1-1111111111-1-1-1-1-1-1-1-111111    linear of order 2
ρ31-1-ii11111-1-1-1-1-i-ii-ii-iii11111    linear of order 4
ρ41-1i-i11111-1-1-1-1ii-ii-ii-i-i11111    linear of order 4
ρ511-1-11ζ52ζ53ζ5ζ54ζ5ζ54ζ53ζ52525355545452531ζ5ζ54ζ52ζ53    linear of order 10
ρ611111ζ53ζ52ζ54ζ5ζ54ζ5ζ52ζ53ζ53ζ52ζ54ζ54ζ5ζ5ζ53ζ521ζ54ζ5ζ53ζ52    linear of order 5
ρ711111ζ52ζ53ζ5ζ54ζ5ζ54ζ53ζ52ζ52ζ53ζ5ζ5ζ54ζ54ζ52ζ531ζ5ζ54ζ52ζ53    linear of order 5
ρ811111ζ54ζ5ζ52ζ53ζ52ζ53ζ5ζ54ζ54ζ5ζ52ζ52ζ53ζ53ζ54ζ51ζ52ζ53ζ54ζ5    linear of order 5
ρ911-1-11ζ5ζ54ζ53ζ52ζ53ζ52ζ54ζ5554535352525541ζ53ζ52ζ5ζ54    linear of order 10
ρ1011-1-11ζ53ζ52ζ54ζ5ζ54ζ5ζ52ζ53535254545553521ζ54ζ5ζ53ζ52    linear of order 10
ρ1111-1-11ζ54ζ5ζ52ζ53ζ52ζ53ζ5ζ54545525253535451ζ52ζ53ζ54ζ5    linear of order 10
ρ1211111ζ5ζ54ζ53ζ52ζ53ζ52ζ54ζ5ζ5ζ54ζ53ζ53ζ52ζ52ζ5ζ541ζ53ζ52ζ5ζ54    linear of order 5
ρ131-1-ii1ζ53ζ52ζ54ζ55455253ζ43ζ53ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53ζ4ζ521ζ54ζ5ζ53ζ52    linear of order 20
ρ141-1-ii1ζ54ζ5ζ52ζ535253554ζ43ζ54ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54ζ4ζ51ζ52ζ53ζ54ζ5    linear of order 20
ρ151-1i-i1ζ54ζ5ζ52ζ535253554ζ4ζ54ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54ζ43ζ51ζ52ζ53ζ54ζ5    linear of order 20
ρ161-1-ii1ζ5ζ54ζ53ζ525352545ζ43ζ5ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5ζ4ζ541ζ53ζ52ζ5ζ54    linear of order 20
ρ171-1i-i1ζ5ζ54ζ53ζ525352545ζ4ζ5ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5ζ43ζ541ζ53ζ52ζ5ζ54    linear of order 20
ρ181-1i-i1ζ52ζ53ζ5ζ545545352ζ4ζ52ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52ζ43ζ531ζ5ζ54ζ52ζ53    linear of order 20
ρ191-1-ii1ζ52ζ53ζ5ζ545545352ζ43ζ52ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52ζ4ζ531ζ5ζ54ζ52ζ53    linear of order 20
ρ201-1i-i1ζ53ζ52ζ54ζ55455253ζ4ζ53ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53ζ43ζ521ζ54ζ5ζ53ζ52    linear of order 20
ρ21400044444000000000000-1-1-1-1-1    orthogonal lifted from F5
ρ22400045455253000000000000-15253545    complex lifted from C5×F5
ρ23400045352545000000000000-15455352    complex lifted from C5×F5
ρ24400045545352000000000000-15352554    complex lifted from C5×F5
ρ25400045253554000000000000-15545253    complex lifted from C5×F5
ρ2620000-5000000000000000000000    orthogonal faithful

Permutation representations of C25⋊C20
On 25 points - transitive group 25T40
Generators in S25
(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)
(2 18 15 14 22 8 20 24 17 23 25 9 12 13 5 19 7 3 10 4)(6 11 21 16)

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

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

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

G:=TransitiveGroup(25,40);

Matrix representation of C25⋊C20 in GL20(ℤ)

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

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

C25⋊C20 in GAP, Magma, Sage, TeX

C_{25}\rtimes C_{20}
% in TeX

G:=Group("C25:C20");
// GroupNames label

G:=SmallGroup(500,18);
// by ID

G=gap.SmallGroup(500,18);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,8803,1208,373,418,5004,1014]);
// Polycyclic

G:=Group<a,b|a^25=b^20=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C25⋊C20 in TeX
Character table of C25⋊C20 in TeX

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