Finite Group DatabasebetaFinite Group Databasebeta| order | 8001 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C9 · Schur multiplier (H2) = C1 |
| direct product | C7×#11765917 |
| derived, Sylow | C127 ⋊ C7×C9 |
| Hall | C7×C127 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C7 | C127⋊C3 |
| commutator, hypocenter | C127 | C7×C9 |
| Frattini | C3 | C7×C127⋊C3 |
| Fitting, socle | C3×C7×C127 | C3 |
| LCS | G ▷ C127 | |
| UCS | C1 ◁ C3×C7 | |
12 conjugacy classes of subgroups.
945 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C7×#11765917, C7×#11765917⋊C2, D7×#11765917, … |
| maximal normal subgroup of | C2×C7×#11765917, C7×#11765917⋊C2, D7×#11765917, … |
| maximal quotient of | C2×C7×#11765917, C7×C22⋊#11765917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8001, 5);