Finite Group DatabasebetaFinite Group Databasebeta| order | 8700 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C5 |
| unambiguous semidirect product | D87⋊(C5⋊D5) |
| Fitting, socle, derived, Hall | C3×C52×C29 ⋊ C22 |
| Hall | C52×C29 ⋊ C2×D3 |
| Hall | C3×C29 ⋊ C2×C5⋊D5 |
| Hall | C3×C52 ⋊ C2×D29 |
| Sylow | C29 ⋊ D3×C5⋊D5 |
| Sylow | C52 ⋊ D3×D29 |
| Sylow | C3 ⋊ C5⋊D5×D29 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C52×C29 | C22 |
| LCS | G ▷ C3×C52×C29 | |
| UCS | C1 | |
160 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,5^2,29|d^a=d^4,e^a=e^4,f^a=f^28,c^b=c^2,d^b=d^4,e^b=e^4588 conjugacy classes — view the full character table.
| maximal subgroup of | D261⋊(C5⋊D5), D87⋊((C5⋊D5)⋊C3), C3×D87⋊(C5⋊D5), … |
| maximal normal subgroup of | D87⋊((C5⋊D5)⋊C3), C3×D87⋊(C5⋊D5) |
| maximal quotient of | D261⋊(C5⋊D5), C3×D87⋊(C5⋊D5), D87⋊(C3⋊(C5⋊D5)), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8700, 127);