Finite Group DatabasebetaFinite Group Databasebeta| order | 8991 |
| automorphism | ? |
| homology | Abelianization (H1) = C34 · Schur multiplier (H2) = C35 |
| unambiguous semidirect product | C37⋊H(5,𝔽3) |
| Fitting | C3×H(3,𝔽3)×C37 ⋊ C3 |
| Sylow | C37 ⋊ H(5,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C33×C37⋊C3 |
| commutator, socle | C3×C37 | C34 |
| Fitting | C3×H(3,𝔽3)×C37 | C3 |
| hypercenter | C3×H(3,𝔽3) | C37⋊C3 |
| hypocenter | C37 | H(5,𝔽3) |
| LCS | G ▷ C3×C37 ▷ C37 | |
| UCS | C1 ◁ C3 ◁ C3×H(3,𝔽3) | |
586 conjugacy classes of subgroups.
Polycyclic presentation:
3^5,37|b^a=be,d^a=de,c^b=ce,f^d=f^26479 conjugacy classes — view the full character table.
| maximal subgroup of | #41006571, #41006572, #41006574, … |
| maximal normal subgroup of | #41006571, #41006572, #41006574, … |
| maximal quotient of | #41007515, #41007516, #41007517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8991, 304);