Finite Group DatabasebetaFinite Group Databasebeta| order | 8991 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C36 |
| unambiguous semidirect product | (C3×C37⋊C3)⋊H(3,𝔽3) |
| Fitting | C3×H(3,𝔽3)×C37 ⋊ C3 |
| Sylow | C37 ⋊ C32⋊H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C32×C37⋊C3 |
| commutator, socle | C32×C37 | C33 |
| Fitting | C3×H(3,𝔽3)×C37 | C3 |
| hypercenter | C3×H(3,𝔽3) | C37⋊C3 |
| hypocenter | C37 | C32⋊H(3,𝔽3) |
| LCS | G ▷ C32×C37 ▷ C37 | |
| UCS | C1 ◁ C32 ◁ C3×H(3,𝔽3) | |
378 conjugacy classes of subgroups.
Polycyclic presentation:
3^5,37|b^a=bd,c^a=ce,f^c=f^26447 conjugacy classes — view the full character table.
| maximal subgroup of | #41005366, #41005367, #41005376, … |
| maximal normal subgroup of | #41005366, #41005367, #41005376, … |
| maximal quotient of | #41005097, #41005122, #41005123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8991, 177);