Finite Group DatabasebetaFinite Group Databasebeta| order | 8470 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊#11925898 |
| Fitting, socle, derived, Hall | C7×C112 ⋊ C2×C5 |
| Hall | C7×#10597244 ⋊ C2 |
| Hall | C11×D77 ⋊ C5 |
| Hall | #10597244 ⋊ D7 |
| Sylow | C112 ⋊ C5×D7 |
| Sylow | C7 ⋊ #11925898 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C112 | C2×C5 |
| LCS | G ▷ C7×C112 | |
| UCS | C1 | |
34 conjugacy classes of subgroups.
Polycyclic presentation:
2,5,7,11^2|c^a=c^6,d^a=d^10,d^b=d^4,e^b=e^5112 conjugacy classes — view the full character table.
| maximal subgroup of | #11925897⋊D7, #11925894⋊D7, D7×#11925898, … |
| maximal normal subgroup of | #11925897⋊D7, #11925894⋊D7, D7×#11925898, … |
| maximal quotient of | C7⋊#14559057, C2×C7⋊#11925898, C3×C7⋊#11925898, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8470, 33);