Finite Group DatabasebetaFinite Group Databasebeta| order | 1500 |
| isoclinism | isoclinic class of C3⋊#92517 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#92517 |
| Fitting, derived, Hall | C3×C125 ⋊ C4 |
| Sylow | C125 ⋊ Dic3 |
| Sylow | C3 ⋊ #92517 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C3×C125 | C4 |
| Frattini | C25 | ΓL(1,𝔽16) |
| socle | C3×C5 | #1035 |
| prime radical, cosocle | C3×D125 | C2 |
| LCS | G ▷ C3×C125 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 375 |
Polycyclic presentation:
2^2,3,5^3|a^2=b,d^5=ef^4,e^5=f,c^a=c^2,d^a=d^2ef,e^a=e^2f,f^a=f^2,d^b=d^4e^4,e^b=e^4f^4,f^b=f^499 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1500, 6);