Finite Group DatabasebetaFinite Group Databasebeta| order | 8540 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊(Dic61⋊C5) |
| unambiguous semidirect product | Dic427⋊C5 |
| derived, Hall | C7×C61 ⋊ C4×C5 |
| Hall | C7×C61⋊C5 ⋊ C4 |
| Hall | Dic427 ⋊ C5 |
| Hall | C61⋊C5 ⋊ Dic7 |
| Sylow | C61 ⋊ C5×Dic7 |
| Sylow | C7 ⋊ Dic61⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C7⋊(D61⋊C5) |
| commutator, hypocenter | C7×C61 | C4×C5 |
| Fitting, socle | C2×C7×C61 | C2×C5 |
| LCS | G ▷ C7×C61 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,5,7,61,2|a^2=e,c^a=c^6,d^a=d^60,d^b=d^9134 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8540, 12);