Finite Group DatabasebetaFinite Group Databasebeta| order | 999 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C9×C37 ⋊ C3 |
| Sylow | C37 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C37⋊C3 |
| commutator, socle, cosocle | C3×C37 | C32 |
| Fitting | C9×C37 | C3 |
| prime radical | C3×C37⋊C3 | C3 |
| hypercenter | C9 | C37⋊C3 |
| hypocenter | C37 | 3−1+2 |
| LCS | G ▷ C3×C37 ▷ C37 | |
| UCS | C1 ◁ C3 ◁ C9 | |
16 conjugacy classes of subgroups.
| transitive degree | μt(G) = 333 |
Polycyclic presentation:
3^3,37|b^3=c,b^a=bc,d^a=d^26119 conjugacy classes — view the full character table.
| maximal subgroup of | #14521981, #14521986, #14521991, … |
| maximal normal subgroup of | #14521981, #14521986, #14521991, … |
| maximal quotient of | C2×#11758605, #15688502, #15688504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(999, 5);