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G = C52:F5  order 500 = 22·53

3rd semidirect product of C52 and F5 acting faithfully

non-abelian, supersoluble, monomial

Aliases: He5:3C4, C52:3F5, He5:C2.2C2, C5.2(C5:F5), SmallGroup(500,23)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — He5 — C52:F5
Chief series
C1 — C5 — C52 — He5 — He5:C2 — C52:F5
Lower central
He5 — C52:F5
Upper central
C1

Generators and relations for C52:F5
 G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, cac-1=ab-1, dad-1=a3b-1, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 481 in 45 conjugacy classes, 11 normal (5 characteristic)
Quotients: C1, C2, C4, F5, C5:F5, C52:F5
C1
25C2
C5
5C5
5C5
5C5
5C5
5C5
5C5
125C4
5D5
5D5
5D5
5D5
5D5
5D5
25C10
C52
C52
C52
C52
C52
C52
25F5
25F5
25F5
25F5
25F5
25F5
25Dic5
5C5xD5
5C5xD5
5C5xD5
5C5xD5
5C5xD5
5C5xD5
He5
5D5.D5
5D5.D5
5D5.D5
5D5.D5
5D5.D5
5D5.D5
He5:C2
C52:F5

Character table of C52:F5

 class 124A4B5A5B5C5D5E5F5G5H10A10B
 size 125125125222020202020205050
ρ111111111111111    trivial
ρ211-1-11111111111    linear of order 2
ρ31-1i-i11111111-1-1    linear of order 4
ρ41-1-ii11111111-1-1    linear of order 4
ρ5400044-1-1-14-1-100    orthogonal lifted from F5
ρ6400044-1-1-1-1-1400    orthogonal lifted from F5
ρ7400044-1-14-1-1-100    orthogonal lifted from F5
ρ8400044-1-1-1-14-100    orthogonal lifted from F5
ρ9400044-14-1-1-1-100    orthogonal lifted from F5
ρ104000444-1-1-1-1-100    orthogonal lifted from F5
ρ1110200-5+5√5/2-5-5√5/2000000-1-√5/2-1+√5/2    orthogonal faithful
ρ1210200-5-5√5/2-5+5√5/2000000-1+√5/2-1-√5/2    orthogonal faithful
ρ1310-200-5-5√5/2-5+5√5/20000001-√5/21+√5/2    symplectic faithful, Schur index 2
ρ1410-200-5+5√5/2-5-5√5/20000001+√5/21-√5/2    symplectic faithful, Schur index 2

Permutation representations of C52:F5
►On 25 points - transitive group 25T33
Generators in S25
(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)

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

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

(1 12 6 24)(2 14 7 25)(3 13 9 22)(4 11 8 21)(5 15 10 23)(17 20)(18 19)
 
G:=sub<Sym(25)| (6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,4,5,2,3)(6,8,10,7,9)(11,12,13,14,15)(16,20,19,18,17)(21,24,22,25,23), (1,20,6,25,14)(2,17,7,24,12)(3,16,9,22,13)(4,19,8,23,15)(5,18,10,21,11), (1,12,6,24)(2,14,7,25)(3,13,9,22)(4,11,8,21)(5,15,10,23)(17,20)(18,19)>;
 
G:=Group( (6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,4,5,2,3)(6,8,10,7,9)(11,12,13,14,15)(16,20,19,18,17)(21,24,22,25,23), (1,20,6,25,14)(2,17,7,24,12)(3,16,9,22,13)(4,19,8,23,15)(5,18,10,21,11), (1,12,6,24)(2,14,7,25)(3,13,9,22)(4,11,8,21)(5,15,10,23)(17,20)(18,19) );
 
G=PermutationGroup([[(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25)], [(1,4,5,2,3),(6,8,10,7,9),(11,12,13,14,15),(16,20,19,18,17),(21,24,22,25,23)], [(1,20,6,25,14),(2,17,7,24,12),(3,16,9,22,13),(4,19,8,23,15),(5,18,10,21,11)], [(1,12,6,24),(2,14,7,25),(3,13,9,22),(4,11,8,21),(5,15,10,23),(17,20),(18,19)]])
 
G:=TransitiveGroup(25,33);
 

Matrix representation of C52:F5 ►in GL10(F41)

344000000000
1000000000
0077000000
003440000000
000040340000
0000770000
0000000100
000000403400
0000000010
0000000001
,
0100000000
403400000000
0001000000
004034000000
0000010000
000040340000
0000000100
000000403400
0000000001
000000004034
,
0000000010
0000000001
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
0000001000
0000000100
,
1000000000
344000000000
0000001000
000000344000
0010000000
003440000000
0000000010
000000003440
0000100000
000034400000

G:=sub<GL(10,GF(41))| [34,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,7,34,0,0,0,0,0,0,0,0,7,40,0,0,0,0,0,0,0,0,0,0,40,7,0,0,0,0,0,0,0,0,34,7,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,34],[0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0],[1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0] >;
 

C52:F5 in GAP, Magma, Sage, TeX 

C_5^2\rtimes F_5
 
% in TeX
 
G:=Group("C5^2:F5");
 
// GroupNames label
 
G:=SmallGroup(500,23);
 
// by ID
 
G=gap.SmallGroup(500,23);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,10,122,127,803,808,613,10004]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a^3*b^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Subgroup lattice of C52:F5 in TeX
Character table of C52:F5 in TeX

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