Finite Group DatabasebetaFinite Group Databasebeta| order | 8001 |
| automorphism | ? |
| homology | Abelianization (H1) = C9 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Hall | C7×C127 ⋊ C9 |
| Sylow | C127 ⋊ C7⋊C9 |
| Sylow | C7 ⋊ #11765919 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C7×C127 | C9 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
3^2,7,127|a^3=b,c^a=c^2,d^a=d^68,d^b=d^107113 conjugacy classes — view the full character table.
| maximal subgroup of | #30630682, #30630684, #30630688, … |
| maximal normal subgroup of | #30630682, #30630684, #30630688, … |
| maximal quotient of | C2×#22547883, C22⋊#22547883 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8001, 12);