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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 >

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
5C5×D5
5C5×D5
5C5×D5
5C5×D5
5C5×D5
5C5×D5
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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