Finite Group DatabasebetaFinite Group Databasebeta| order | 8991 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C81 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C37⋊C3)⋊C81 |
| derived, Sylow | C37 ⋊ C3×C81 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C27 | #66526 |
| commutator, hypocenter | C37 | C3×C81 |
| Frattini | C9 | C3×#66526 |
| Fitting | C27×C37 | C9 |
| socle | C3×C37 | C3×C27 |
| LCS | G ▷ C37 | |
| UCS | C1 ◁ C27 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
3^5,37|a^3=c,c^3=d,d^3=e,f^a=f^16,f^b=f^10,f^c=f^26351 conjugacy classes — view the full character table.
| maximal subgroup of | C81×#66526, #41008344, #41008345, … |
| maximal normal subgroup of | C81×#66526, #41008344, #41008345, … |
| maximal quotient of | #41008343, #41008359, #41008420, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8991, 342);