metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C25⋊C4, C5.F5, D25.C2, SmallGroup(100,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C25⋊C4 |
Generators and relations for C25⋊C4
G = < a,b | a25=b4=1, bab-1=a18 >
Character table of C25⋊C4
| class | 1 | 2 | 4A | 4B | 5 | 25A | 25B | 25C | 25D | 25E | |
| 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 | 4 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ6 | 4 | 0 | 0 | 0 | -1 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2524+ζ2518+ζ257+ζ25 | ζ2522+ζ2521+ζ254+ζ253 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | orthogonal faithful |
| ρ7 | 4 | 0 | 0 | 0 | -1 | ζ2524+ζ2518+ζ257+ζ25 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2519+ζ2517+ζ258+ζ256 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | orthogonal faithful |
| ρ8 | 4 | 0 | 0 | 0 | -1 | ζ2519+ζ2517+ζ258+ζ256 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | orthogonal faithful |
| ρ9 | 4 | 0 | 0 | 0 | -1 | ζ2522+ζ2521+ζ254+ζ253 | ζ2519+ζ2517+ζ258+ζ256 | ζ2524+ζ2518+ζ257+ζ25 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | orthogonal faithful |
| ρ10 | 4 | 0 | 0 | 0 | -1 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2522+ζ2521+ζ254+ζ253 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | orthogonal faithful |
►On 25 points - transitive group 25T8
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)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 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), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13)>;
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), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13) );
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)], [(2,8,25,19),(3,15,24,12),(4,22,23,5),(6,11,21,16),(7,18,20,9),(10,14,17,13)]])
G:=TransitiveGroup(25,8);
C25⋊C4 is a maximal subgroup of
C75⋊C4 C125⋊C4 C25⋊C20 D25.D5 C25⋊F5 C25⋊2F5
C25⋊C4 is a maximal quotient of C25⋊C8 C75⋊C4 C125⋊C4 D25.D5 C25⋊F5 C25⋊2F5
Matrix representation of C25⋊C4 ►in GL4(𝔽7) generated by
| 5 | 2 | 5 | 5 |
| 6 | 0 | 2 | 3 |
| 5 | 4 | 5 | 2 |
| 6 | 1 | 1 | 6 |
| 0 | 1 | 3 | 0 |
| 0 | 6 | 1 | 2 |
| 2 | 4 | 3 | 2 |
| 2 | 6 | 3 | 5 |
G:=sub<GL(4,GF(7))| [5,6,5,6,2,0,4,1,5,2,5,1,5,3,2,6],[0,0,2,2,1,6,4,6,3,1,3,3,0,2,2,5] >;
C25⋊C4 in GAP, Magma, Sage, TeX
C_{25}\rtimes C_4 % in TeX
G:=Group("C25:C4"); // GroupNames label
G:=SmallGroup(100,3);
// by ID
G=gap.SmallGroup(100,3);
# by ID
G:=PCGroup([4,-2,-2,-5,-5,8,338,582,70,643,647]);
// Polycyclic
G:=Group<a,b|a^25=b^4=1,b*a*b^-1=a^18>;
// generators/relations
Export
Subgroup lattice of C25⋊C4 in TeX
Character table of C25⋊C4 in TeX