direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C5×F5, C5⋊C20, D5.C10, C52⋊1C4, (C5×D5).1C2, SmallGroup(100,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C5×F5 |
Generators and relations for C5×F5
G = < a,b,c | a5=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Character table of C5×F5
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 5I | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | |
| size | 1 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | -i | i | -i | i | -i | i | linear of order 4 |
| ρ4 | 1 | -1 | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | i | -i | i | -i | i | -i | linear of order 4 |
| ρ5 | 1 | 1 | -1 | -1 | ζ52 | ζ53 | ζ5 | ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | -ζ53 | -ζ53 | -ζ5 | -ζ5 | -ζ54 | -ζ54 | -ζ52 | -ζ52 | linear of order 10 |
| ρ6 | 1 | 1 | 1 | 1 | ζ5 | ζ54 | ζ53 | ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | ζ54 | ζ54 | ζ53 | ζ53 | ζ52 | ζ52 | ζ5 | ζ5 | linear of order 5 |
| ρ7 | 1 | 1 | 1 | 1 | ζ54 | ζ5 | ζ52 | ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | 1 | ζ52 | ζ54 | ζ5 | ζ53 | ζ5 | ζ5 | ζ52 | ζ52 | ζ53 | ζ53 | ζ54 | ζ54 | linear of order 5 |
| ρ8 | 1 | 1 | -1 | -1 | ζ5 | ζ54 | ζ53 | ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | 1 | ζ53 | ζ5 | ζ54 | ζ52 | -ζ54 | -ζ54 | -ζ53 | -ζ53 | -ζ52 | -ζ52 | -ζ5 | -ζ5 | linear of order 10 |
| ρ9 | 1 | 1 | 1 | 1 | ζ52 | ζ53 | ζ5 | ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | 1 | ζ5 | ζ52 | ζ53 | ζ54 | ζ53 | ζ53 | ζ5 | ζ5 | ζ54 | ζ54 | ζ52 | ζ52 | linear of order 5 |
| ρ10 | 1 | 1 | 1 | 1 | ζ53 | ζ52 | ζ54 | ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | ζ52 | ζ52 | ζ54 | ζ54 | ζ5 | ζ5 | ζ53 | ζ53 | linear of order 5 |
| ρ11 | 1 | 1 | -1 | -1 | ζ53 | ζ52 | ζ54 | ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | 1 | ζ54 | ζ53 | ζ52 | ζ5 | -ζ52 | -ζ52 | -ζ54 | -ζ54 | -ζ5 | -ζ5 | -ζ53 | -ζ53 | linear of order 10 |
| ρ12 | 1 | 1 | -1 | -1 | ζ54 | ζ5 | ζ52 | ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | 1 | ζ52 | ζ54 | ζ5 | ζ53 | -ζ5 | -ζ5 | -ζ52 | -ζ52 | -ζ53 | -ζ53 | -ζ54 | -ζ54 | linear of order 10 |
| ρ13 | 1 | -1 | i | -i | ζ5 | ζ54 | ζ53 | ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | 1 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | ζ43ζ54 | ζ4ζ54 | ζ43ζ53 | ζ4ζ53 | ζ43ζ52 | ζ4ζ52 | ζ43ζ5 | ζ4ζ5 | linear of order 20 |
| ρ14 | 1 | -1 | i | -i | ζ52 | ζ53 | ζ5 | ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | 1 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | ζ43ζ53 | ζ4ζ53 | ζ43ζ5 | ζ4ζ5 | ζ43ζ54 | ζ4ζ54 | ζ43ζ52 | ζ4ζ52 | linear of order 20 |
| ρ15 | 1 | -1 | -i | i | ζ53 | ζ52 | ζ54 | ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | 1 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | ζ4ζ52 | ζ43ζ52 | ζ4ζ54 | ζ43ζ54 | ζ4ζ5 | ζ43ζ5 | ζ4ζ53 | ζ43ζ53 | linear of order 20 |
| ρ16 | 1 | -1 | -i | i | ζ5 | ζ54 | ζ53 | ζ52 | ζ53 | ζ54 | ζ52 | ζ5 | 1 | -ζ53 | -ζ5 | -ζ54 | -ζ52 | ζ4ζ54 | ζ43ζ54 | ζ4ζ53 | ζ43ζ53 | ζ4ζ52 | ζ43ζ52 | ζ4ζ5 | ζ43ζ5 | linear of order 20 |
| ρ17 | 1 | -1 | -i | i | ζ54 | ζ5 | ζ52 | ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | 1 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | ζ4ζ5 | ζ43ζ5 | ζ4ζ52 | ζ43ζ52 | ζ4ζ53 | ζ43ζ53 | ζ4ζ54 | ζ43ζ54 | linear of order 20 |
| ρ18 | 1 | -1 | -i | i | ζ52 | ζ53 | ζ5 | ζ54 | ζ5 | ζ53 | ζ54 | ζ52 | 1 | -ζ5 | -ζ52 | -ζ53 | -ζ54 | ζ4ζ53 | ζ43ζ53 | ζ4ζ5 | ζ43ζ5 | ζ4ζ54 | ζ43ζ54 | ζ4ζ52 | ζ43ζ52 | linear of order 20 |
| ρ19 | 1 | -1 | i | -i | ζ54 | ζ5 | ζ52 | ζ53 | ζ52 | ζ5 | ζ53 | ζ54 | 1 | -ζ52 | -ζ54 | -ζ5 | -ζ53 | ζ43ζ5 | ζ4ζ5 | ζ43ζ52 | ζ4ζ52 | ζ43ζ53 | ζ4ζ53 | ζ43ζ54 | ζ4ζ54 | linear of order 20 |
| ρ20 | 1 | -1 | i | -i | ζ53 | ζ52 | ζ54 | ζ5 | ζ54 | ζ52 | ζ5 | ζ53 | 1 | -ζ54 | -ζ53 | -ζ52 | -ζ5 | ζ43ζ52 | ζ4ζ52 | ζ43ζ54 | ζ4ζ54 | ζ43ζ5 | ζ4ζ5 | ζ43ζ53 | ζ4ζ53 | linear of order 20 |
| ρ21 | 4 | 0 | 0 | 0 | 4 | 4 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F5 |
| ρ22 | 4 | 0 | 0 | 0 | 4ζ54 | 4ζ5 | 4ζ52 | 4ζ53 | -ζ52 | -ζ5 | -ζ53 | -ζ54 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ23 | 4 | 0 | 0 | 0 | 4ζ53 | 4ζ52 | 4ζ54 | 4ζ5 | -ζ54 | -ζ52 | -ζ5 | -ζ53 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 4 | 0 | 0 | 0 | 4ζ5 | 4ζ54 | 4ζ53 | 4ζ52 | -ζ53 | -ζ54 | -ζ52 | -ζ5 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ25 | 4 | 0 | 0 | 0 | 4ζ52 | 4ζ53 | 4ζ5 | 4ζ54 | -ζ5 | -ζ53 | -ζ54 | -ζ52 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 20 points - transitive group 20T29
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 2 3 4 5)(6 10 9 8 7)(11 13 15 12 14)(16 19 17 20 18)
(1 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)
G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,2,3,4,5)(6,10,9,8,7)(11,13,15,12,14)(16,19,17,20,18), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)>;
G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,2,3,4,5)(6,10,9,8,7)(11,13,15,12,14)(16,19,17,20,18), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15) );
G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,2,3,4,5),(6,10,9,8,7),(11,13,15,12,14),(16,19,17,20,18)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15)]])
G:=TransitiveGroup(20,29);
►On 25 points - transitive group 25T7
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 19 12 9 25)(2 20 13 10 21)(3 16 14 6 22)(4 17 15 7 23)(5 18 11 8 24)
(6 16 14 22)(7 17 15 23)(8 18 11 24)(9 19 12 25)(10 20 13 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), (1,19,12,9,25)(2,20,13,10,21)(3,16,14,6,22)(4,17,15,7,23)(5,18,11,8,24), (6,16,14,22)(7,17,15,23)(8,18,11,24)(9,19,12,25)(10,20,13,21)>;
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), (1,19,12,9,25)(2,20,13,10,21)(3,16,14,6,22)(4,17,15,7,23)(5,18,11,8,24), (6,16,14,22)(7,17,15,23)(8,18,11,24)(9,19,12,25)(10,20,13,21) );
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)], [(1,19,12,9,25),(2,20,13,10,21),(3,16,14,6,22),(4,17,15,7,23),(5,18,11,8,24)], [(6,16,14,22),(7,17,15,23),(8,18,11,24),(9,19,12,25),(10,20,13,21)]])
G:=TransitiveGroup(25,7);
C5×F5 is a maximal subgroup of
C52⋊C20 C25⋊C20 He5⋊4C4
C5×F5 is a maximal quotient of C52⋊C20 C25⋊C20
Matrix representation of C5×F5 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 18 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 6 | 25 | 10 | 10 |
| 0 | 0 | 1 | 0 |
| 40 | 40 | 40 | 15 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[18,0,0,6,0,16,0,25,0,0,37,10,0,0,0,10],[0,40,0,0,0,40,1,0,1,40,0,0,0,15,0,1] >;
C5×F5 in GAP, Magma, Sage, TeX
C_5\times F_5
% in TeX
G:=Group("C5xF5"); // GroupNames label
G:=SmallGroup(100,9);
// by ID
G=gap.SmallGroup(100,9);
# by ID
G:=PCGroup([4,-2,-5,-2,-5,40,643,139]);
// Polycyclic
G:=Group<a,b,c|a^5=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
Export
Subgroup lattice of C5×F5 in TeX
Character table of C5×F5 in TeX