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G = C52⋊C20order 500 = 22·53

The semidirect product of C52 and C20 acting faithfully

metabelian, supersoluble, monomial

Aliases: C52⋊C20, He51C4, C521F5, C5⋊D5.C10, C5⋊F5⋊C5, C5.2(C5×F5), C52⋊C10.1C2, SmallGroup(500,17)

Series: Derived Chief Lower central Upper central

C1C52 — C52⋊C20
C1C5C52C5⋊D5C52⋊C10 — C52⋊C20
C52 — C52⋊C20
C1

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

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

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
ρ511-1-11ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ531ζ54ζ53ζ52ζ552525454555353    linear of order 10
ρ611111ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ51ζ53ζ5ζ54ζ52ζ54ζ54ζ53ζ53ζ52ζ52ζ5ζ5    linear of order 5
ρ711111ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ541ζ52ζ54ζ5ζ53ζ5ζ5ζ52ζ52ζ53ζ53ζ54ζ54    linear of order 5
ρ811111ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ521ζ5ζ52ζ53ζ54ζ53ζ53ζ5ζ5ζ54ζ54ζ52ζ52    linear of order 5
ρ911-1-11ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ541ζ52ζ54ζ5ζ5355525253535454    linear of order 10
ρ1011-1-11ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ51ζ53ζ5ζ54ζ5254545353525255    linear of order 10
ρ1111-1-11ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ521ζ5ζ52ζ53ζ5453535554545252    linear of order 10
ρ1211111ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ531ζ54ζ53ζ52ζ5ζ52ζ52ζ54ζ54ζ5ζ5ζ53ζ53    linear of order 5
ρ131-1-ii1ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ5215525354ζ4ζ53ζ43ζ53ζ4ζ5ζ43ζ5ζ4ζ54ζ43ζ54ζ4ζ52ζ43ζ52    linear of order 20
ρ141-1i-i1ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ515355452ζ43ζ54ζ4ζ54ζ43ζ53ζ4ζ53ζ43ζ52ζ4ζ52ζ43ζ5ζ4ζ5    linear of order 20
ρ151-1i-i1ζ52ζ53ζ5ζ54ζ5ζ53ζ54ζ5215525354ζ43ζ53ζ4ζ53ζ43ζ5ζ4ζ5ζ43ζ54ζ4ζ54ζ43ζ52ζ4ζ52    linear of order 20
ρ161-1-ii1ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ5415254553ζ4ζ5ζ43ζ5ζ4ζ52ζ43ζ52ζ4ζ53ζ43ζ53ζ4ζ54ζ43ζ54    linear of order 20
ρ171-1-ii1ζ5ζ54ζ53ζ52ζ53ζ54ζ52ζ515355452ζ4ζ54ζ43ζ54ζ4ζ53ζ43ζ53ζ4ζ52ζ43ζ52ζ4ζ5ζ43ζ5    linear of order 20
ρ181-1i-i1ζ54ζ5ζ52ζ53ζ52ζ5ζ53ζ5415254553ζ43ζ5ζ4ζ5ζ43ζ52ζ4ζ52ζ43ζ53ζ4ζ53ζ43ζ54ζ4ζ54    linear of order 20
ρ191-1-ii1ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ5315453525ζ4ζ52ζ43ζ52ζ4ζ54ζ43ζ54ζ4ζ5ζ43ζ5ζ4ζ53ζ43ζ53    linear of order 20
ρ201-1i-i1ζ53ζ52ζ54ζ5ζ54ζ52ζ5ζ5315453525ζ43ζ52ζ4ζ52ζ43ζ54ζ4ζ54ζ43ζ5ζ4ζ5ζ43ζ53ζ4ζ53    linear of order 20
ρ21400044444-1-1-1-1-1000000000000    orthogonal lifted from F5
ρ224000454552535255354-1000000000000    complex lifted from C5×F5
ρ234000453525455452553-1000000000000    complex lifted from C5×F5
ρ244000455453525354525-1000000000000    complex lifted from C5×F5
ρ254000452535545535452-1000000000000    complex lifted from C5×F5
ρ2620000-5000000000000000000000    orthogonal faithful

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