metabelian, supersoluble, monomial, A-group
Aliases: C5⋊1F5, C52⋊3C4, C5⋊D5.1C2, SmallGroup(100,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C5⋊F5 |
Generators and relations for C5⋊F5
G = < a,b,c | a5=b5=c4=1, ab=ba, cac-1=a3, cbc-1=b3 >
Character table of C5⋊F5
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | |
| size | 1 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ4 | 1 | -1 | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ5 | 4 | 0 | 0 | 0 | -1 | -1 | 4 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ6 | 4 | 0 | 0 | 0 | -1 | 4 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ7 | 4 | 0 | 0 | 0 | -1 | -1 | -1 | 4 | -1 | -1 | orthogonal lifted from F5 |
| ρ8 | 4 | 0 | 0 | 0 | 4 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ9 | 4 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 4 | orthogonal lifted from F5 |
| ρ10 | 4 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 4 | -1 | orthogonal lifted from 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 9 13 16)(2 24 10 14 17)(3 25 6 15 18)(4 21 7 11 19)(5 22 8 12 20)
(2 3 5 4)(6 20 11 24)(7 17 15 22)(8 19 14 25)(9 16 13 23)(10 18 12 21)
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), (1,23,9,13,16)(2,24,10,14,17)(3,25,6,15,18)(4,21,7,11,19)(5,22,8,12,20), (2,3,5,4)(6,20,11,24)(7,17,15,22)(8,19,14,25)(9,16,13,23)(10,18,12,21)>;
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), (1,23,9,13,16)(2,24,10,14,17)(3,25,6,15,18)(4,21,7,11,19)(5,22,8,12,20), (2,3,5,4)(6,20,11,24)(7,17,15,22)(8,19,14,25)(9,16,13,23)(10,18,12,21) );
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)], [(1,23,9,13,16),(2,24,10,14,17),(3,25,6,15,18),(4,21,7,11,19),(5,22,8,12,20)], [(2,3,5,4),(6,20,11,24),(7,17,15,22),(8,19,14,25),(9,16,13,23),(10,18,12,21)]])
G:=TransitiveGroup(25,9);
C5⋊F5 is a maximal subgroup of
C52⋊C8 D5⋊F5 C52⋊C12 C15⋊F5 C52⋊C20 C25⋊F5 C53⋊7C4 C53⋊8C4 C53⋊9C4
C5⋊F5 is a maximal quotient of
C52⋊4C8 C15⋊F5 C25⋊F5 C52⋊F5 C53⋊7C4 C53⋊8C4 C53⋊9C4
Matrix representation of C5⋊F5 ►in GL8(ℤ)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 |
| 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 |
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