Finite Group DatabasebetaFinite Group Databasebeta| order | 1500 |
| isoclinism | isoclinic class of C3⋊(#1043⋊C5) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(#1043⋊C5) |
| unambiguous semidirect product | (C3⋊#1043)⋊C5 |
| Fitting, Hall | C3×H(3,𝔽5) ⋊ C4 |
| derived | C3×C52 ⋊ C4×C5 |
| Sylow | H(3,𝔽5) ⋊ Dic3 |
| Sylow | C3 ⋊ #1043⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C3×C52 | C4×C5 |
| Frattini | C5 | C5×ΓL(1,𝔽16) |
| Fitting | C3×H(3,𝔽5) | C4 |
| socle | C3×C5 | C5×AGL(1,𝔽5) |
| prime radical | C3×(C5⋊D5)⋊C5 | C2 |
| cosocle | C3×C5⋊D5 | C2×C5 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 75 |
Polycyclic presentation:
2,5,2,3,5^2|a^2=c,d^a=d^2,e^a=e^2,f^a=f^2,e^b=ef,e^c=e^4,f^c=f^448 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1500, 53);