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G = C5⋊F5order 100 = 22·52

1st semidirect product of C5 and F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, A-group

Aliases: C51F5, C523C4, C5⋊D5.1C2, SmallGroup(100,11)

Series: Derived Chief Lower central Upper central

C1C52 — C5⋊F5
C1C5C52C5⋊D5 — C5⋊F5
C52 — C5⋊F5
C1

Generators and relations for C5⋊F5
 G = < a,b,c | a5=b5=c4=1, ab=ba, cac-1=a3, cbc-1=b3 >

25C2
25C4
5D5
5D5
5D5
5D5
5D5
5D5
5F5
5F5
5F5
5F5
5F5
5F5

Character table of C5⋊F5

 class 124A4B5A5B5C5D5E5F
 size 1252525444444
ρ11111111111    trivial
ρ211-1-1111111    linear of order 2
ρ31-1-ii111111    linear of order 4
ρ41-1i-i111111    linear of order 4
ρ54000-1-14-1-1-1    orthogonal lifted from F5
ρ64000-14-1-1-1-1    orthogonal lifted from F5
ρ74000-1-1-14-1-1    orthogonal lifted from F5
ρ840004-1-1-1-1-1    orthogonal lifted from F5
ρ94000-1-1-1-1-14    orthogonal lifted from F5
ρ104000-1-1-1-14-1    orthogonal lifted from F5

Permutation representations of C5⋊F5
On 25 points - transitive group 25T9
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)
(1 23 13 20 10)(2 24 14 16 6)(3 25 15 17 7)(4 21 11 18 8)(5 22 12 19 9)
(2 3 5 4)(6 17 22 11)(7 19 21 14)(8 16 25 12)(9 18 24 15)(10 20 23 13)

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

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

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

G:=TransitiveGroup(25,9);

C5⋊F5 is a maximal subgroup of
C52⋊C8  D5⋊F5  C52⋊C12  C15⋊F5  C52⋊C20  C25⋊F5  C537C4  C538C4  C539C4
C5⋊F5 is a maximal quotient of
C524C8  C15⋊F5  C25⋊F5  C52⋊F5  C537C4  C538C4  C539C4

Matrix representation of C5⋊F5 in GL8(ℤ)

00100000
00010000
-1-1-1-10000
10000000
00000100
00000010
00000001
0000-1-1-1-1
,
-1-1-1-10000
10000000
01000000
00100000
0000-1-1-1-1
00001000
00000100
00000010
,
10000000
00010000
01000000
-1-1-1-10000
00000-1-10
00000111
000000-1-1
0000-1-100

G:=sub<GL(8,Integers())| [0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1],[-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,0,0,1,0,0,0,0,-1,0,0,0],[1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,1,0,-1,0,0,0,0,-1,1,-1,0,0,0,0,0,0,1,-1,0] >;

C5⋊F5 in GAP, Magma, Sage, TeX

C_5\rtimes F_5
% in TeX

G:=Group("C5:F5");
// GroupNames label

G:=SmallGroup(100,11);
// by ID

G=gap.SmallGroup(100,11);
# by ID

G:=PCGroup([4,-2,-2,-5,-5,8,98,102,643,647]);
// Polycyclic

G:=Group<a,b,c|a^5=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^3,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C5⋊F5 in TeX
Character table of C5⋊F5 in TeX

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