Finite Group DatabasebetaFinite Group Databasebeta| order | 8001 |
| automorphism | ? |
| homology | Abelianization (H1) = C7×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #11765919⋊C7 |
| Fitting, socle, derived, Sylow | C127 ⋊ C7×C9 |
| Hall | #11765919 ⋊ C7 |
| Hall | C127⋊C7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C127 | C7×C9 |
| LCS | G ▷ C127 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
Polycyclic presentation:
3,7,3,127|a^3=c,d^a=d^68,d^b=d^4,d^c=d^10765 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽127), C2×#11765919⋊C7, C22⋊(#11765919⋊C7) |
| maximal normal subgroup of | AGL(1,𝔽127), C2×#11765919⋊C7 |
| maximal quotient of | C2×#11765919⋊C7, C22⋊(#11765919⋊C7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8001, 9);