direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×C25⋊C4, C25⋊C12, C75⋊2C4, D25.C6, C15.2F5, C5.(C3×F5), (C3×D25).2C2, SmallGroup(300,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C25 — C3×C25⋊C4 |
Generators and relations for C3×C25⋊C4
G = < a,b,c | a3=b25=c4=1, ab=ba, ac=ca, cbc-1=b18 >
Character table of C3×C25⋊C4
| class | 1 | 2 | 3A | 3B | 4A | 4B | 5 | 6A | 6B | 12A | 12B | 12C | 12D | 15A | 15B | 25A | 25B | 25C | 25D | 25E | 75A | 75B | 75C | 75D | 75E | 75F | 75G | 75H | 75I | 75J | |
| size | 1 | 25 | 1 | 1 | 25 | 25 | 4 | 25 | 25 | 25 | 25 | 25 | 25 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | ζ3 | ζ32 | -1 | -1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ6 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 6 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 3 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | -1 | -1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ65 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 6 |
| ρ7 | 1 | -1 | 1 | 1 | i | -i | 1 | -1 | -1 | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | 1 | 1 | -i | i | 1 | -1 | -1 | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | -i | i | 1 | ζ65 | ζ6 | ζ43ζ32 | ζ4ζ3 | ζ43ζ3 | ζ4ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 12 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | i | -i | 1 | ζ6 | ζ65 | ζ4ζ3 | ζ43ζ32 | ζ4ζ32 | ζ43ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 12 |
| ρ11 | 1 | -1 | ζ32 | ζ3 | -i | i | 1 | ζ6 | ζ65 | ζ43ζ3 | ζ4ζ32 | ζ43ζ32 | ζ4ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | linear of order 12 |
| ρ12 | 1 | -1 | ζ3 | ζ32 | i | -i | 1 | ζ65 | ζ6 | ζ4ζ32 | ζ43ζ3 | ζ4ζ3 | ζ43ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | linear of order 12 |
| ρ13 | 4 | 0 | 4 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 4 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ14 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2519+ζ2517+ζ258+ζ256 | orthogonal lifted from C25⋊C4 |
| ρ15 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2522+ζ2521+ζ254+ζ253 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2523+ζ2514+ζ2511+ζ252 | orthogonal lifted from C25⋊C4 |
| ρ16 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2524+ζ2518+ζ257+ζ25 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2516+ζ2513+ζ2512+ζ259 | orthogonal lifted from C25⋊C4 |
| ρ17 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2519+ζ2517+ζ258+ζ256 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2522+ζ2521+ζ254+ζ253 | orthogonal lifted from C25⋊C4 |
| ρ18 | 4 | 0 | 4 | 4 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2524+ζ2518+ζ257+ζ25 | orthogonal lifted from C25⋊C4 |
| ρ19 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2-2√-3 | -2+2√-3 | -1 | -1 | -1 | -1 | -1 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | complex lifted from C3×F5 |
| ρ20 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2+2√-3 | -2-2√-3 | -1 | -1 | -1 | -1 | -1 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | complex lifted from C3×F5 |
| ρ21 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | complex faithful |
| ρ22 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | complex faithful |
| ρ23 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | complex faithful |
| ρ24 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | complex faithful |
| ρ25 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | complex faithful |
| ρ26 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | complex faithful |
| ρ27 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | complex faithful |
| ρ28 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | complex faithful |
| ρ29 | 4 | 0 | -2+2√-3 | -2-2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ2519+ζ2517+ζ258+ζ256 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | complex faithful |
| ρ30 | 4 | 0 | -2-2√-3 | -2+2√-3 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ2519+ζ2517+ζ258+ζ256 | ζ2523+ζ2514+ζ2511+ζ252 | ζ2516+ζ2513+ζ2512+ζ259 | ζ2522+ζ2521+ζ254+ζ253 | ζ2524+ζ2518+ζ257+ζ25 | ζ32ζ2516+ζ32ζ2513+ζ32ζ2512+ζ32ζ259 | ζ3ζ2523+ζ3ζ2514+ζ3ζ2511+ζ3ζ252 | ζ3ζ2516+ζ3ζ2513+ζ3ζ2512+ζ3ζ259 | ζ32ζ2522+ζ32ζ2521+ζ32ζ254+ζ32ζ253 | ζ3ζ2522+ζ3ζ2521+ζ3ζ254+ζ3ζ253 | ζ3ζ2524+ζ3ζ2518+ζ3ζ257+ζ3ζ25 | ζ32ζ2524+ζ32ζ2518+ζ32ζ257+ζ32ζ25 | ζ32ζ2519+ζ32ζ2517+ζ32ζ258+ζ32ζ256 | ζ32ζ2523+ζ32ζ2514+ζ32ζ2511+ζ32ζ252 | ζ3ζ2519+ζ3ζ2517+ζ3ζ258+ζ3ζ256 | complex faithful |
►On 75 points
Generators in S75
(1 68 45)(2 69 46)(3 70 47)(4 71 48)(5 72 49)(6 73 50)(7 74 26)(8 75 27)(9 51 28)(10 52 29)(11 53 30)(12 54 31)(13 55 32)(14 56 33)(15 57 34)(16 58 35)(17 59 36)(18 60 37)(19 61 38)(20 62 39)(21 63 40)(22 64 41)(23 65 42)(24 66 43)(25 67 44)
(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)(26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75)
(2 8 25 19)(3 15 24 12)(4 22 23 5)(6 11 21 16)(7 18 20 9)(10 14 17 13)(26 37 39 28)(27 44 38 46)(29 33 36 32)(30 40 35 50)(31 47 34 43)(41 42 49 48)(51 74 60 62)(52 56 59 55)(53 63 58 73)(54 70 57 66)(61 69 75 67)(64 65 72 71)
G:=sub<Sym(75)| (1,68,45)(2,69,46)(3,70,47)(4,71,48)(5,72,49)(6,73,50)(7,74,26)(8,75,27)(9,51,28)(10,52,29)(11,53,30)(12,54,31)(13,55,32)(14,56,33)(15,57,34)(16,58,35)(17,59,36)(18,60,37)(19,61,38)(20,62,39)(21,63,40)(22,64,41)(23,65,42)(24,66,43)(25,67,44), (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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13)(26,37,39,28)(27,44,38,46)(29,33,36,32)(30,40,35,50)(31,47,34,43)(41,42,49,48)(51,74,60,62)(52,56,59,55)(53,63,58,73)(54,70,57,66)(61,69,75,67)(64,65,72,71)>;
G:=Group( (1,68,45)(2,69,46)(3,70,47)(4,71,48)(5,72,49)(6,73,50)(7,74,26)(8,75,27)(9,51,28)(10,52,29)(11,53,30)(12,54,31)(13,55,32)(14,56,33)(15,57,34)(16,58,35)(17,59,36)(18,60,37)(19,61,38)(20,62,39)(21,63,40)(22,64,41)(23,65,42)(24,66,43)(25,67,44), (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)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75), (2,8,25,19)(3,15,24,12)(4,22,23,5)(6,11,21,16)(7,18,20,9)(10,14,17,13)(26,37,39,28)(27,44,38,46)(29,33,36,32)(30,40,35,50)(31,47,34,43)(41,42,49,48)(51,74,60,62)(52,56,59,55)(53,63,58,73)(54,70,57,66)(61,69,75,67)(64,65,72,71) );
G=PermutationGroup([[(1,68,45),(2,69,46),(3,70,47),(4,71,48),(5,72,49),(6,73,50),(7,74,26),(8,75,27),(9,51,28),(10,52,29),(11,53,30),(12,54,31),(13,55,32),(14,56,33),(15,57,34),(16,58,35),(17,59,36),(18,60,37),(19,61,38),(20,62,39),(21,63,40),(22,64,41),(23,65,42),(24,66,43),(25,67,44)], [(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),(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75)], [(2,8,25,19),(3,15,24,12),(4,22,23,5),(6,11,21,16),(7,18,20,9),(10,14,17,13),(26,37,39,28),(27,44,38,46),(29,33,36,32),(30,40,35,50),(31,47,34,43),(41,42,49,48),(51,74,60,62),(52,56,59,55),(53,63,58,73),(54,70,57,66),(61,69,75,67),(64,65,72,71)]])
Matrix representation of C3×C25⋊C4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 6 | 1 | 6 |
| 5 | 5 | 3 | 6 |
| 1 | 0 | 6 | 0 |
| 4 | 2 | 1 | 6 |
| 1 | 5 | 0 | 0 |
| 1 | 6 | 0 | 0 |
| 4 | 2 | 1 | 5 |
| 2 | 3 | 0 | 6 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,5,1,4,6,5,0,2,1,3,6,1,6,6,0,6],[1,1,4,2,5,6,2,3,0,0,1,0,0,0,5,6] >;
C3×C25⋊C4 in GAP, Magma, Sage, TeX
C_3\times C_{25}\rtimes C_4 % in TeX
G:=Group("C3xC25:C4"); // GroupNames label
G:=SmallGroup(300,5);
// by ID
G=gap.SmallGroup(300,5);
# by ID
G:=PCGroup([5,-2,-3,-2,-5,-5,30,1683,973,118,3004,1014]);
// Polycyclic
G:=Group<a,b,c|a^3=b^25=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^18>;
// generators/relations
Export
Subgroup lattice of C3×C25⋊C4 in TeX
Character table of C3×C25⋊C4 in TeX