Finite Group DatabasebetaFinite Group Databasebeta| order | 999 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C32⋊(C37⋊C3) |
| unambiguous semidirect product | C37⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×C37⋊C3)⋊C3 |
| Fitting | C32×C37 ⋊ C3 |
| Sylow | C37 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C37⋊C3 |
| commutator, socle, cosocle | C3×C37 | C32 |
| Fitting | C32×C37 | C3 |
| hypercenter | C32 | C37⋊C3 |
| hypocenter | C37 | H(3,𝔽3) |
| LCS | G ▷ C3×C37 ▷ C37 | |
| UCS | C1 ◁ C3 ◁ C32 | |
22 conjugacy classes of subgroups.
| transitive degree | μt(G) = 333 |
Polycyclic presentation:
3^3,37|b^a=bc,d^a=d^26119 conjugacy classes — view the full character table.
| maximal subgroup of | D37⋊H(3,𝔽3), (C3⋊D3)⋊(C37⋊C3), #14521994, … |
| maximal normal subgroup of | D37⋊H(3,𝔽3), (C3⋊D3)⋊(C37⋊C3), #14521994, … |
| maximal quotient of | C2×C32⋊(C37⋊C3), #15688502, C32⋊#66524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(999, 9);