Finite Group DatabasebetaFinite Group Databasebeta| order | 8470 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊(D121⋊C5) |
| unambiguous semidirect product | (C121⋊C5)⋊D7 |
| unambiguous semidirect product | D847⋊C5 |
| Fitting, derived, Hall | C7×C121 ⋊ C2×C5 |
| Hall | C7×C121⋊C5 ⋊ C2 |
| Hall | D847 ⋊ C5 |
| Hall | C121⋊C5 ⋊ D7 |
| Sylow | C121 ⋊ C5×D7 |
| Sylow | C7 ⋊ D121⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C7×C121 | C2×C5 |
| Frattini | C11 | C7⋊AGL(1,𝔽11) |
| socle | C7×C11 | AGL(1,𝔽11) |
| LCS | G ▷ C7×C121 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
Polycyclic presentation:
2,5,7,11^2|d^11=e^10,c^a=c^6,d^a=d^10e,e^a=e^10,d^b=d^4e^4,e^b=e^4109 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8470, 4);