Finite Group DatabasebetaFinite Group Databasebeta| order | 999 |
| isoclinism | isoclinic class of C37⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C9×C37⋊C3 |
| Fitting | C9×C37 ⋊ C3 |
| derived, Sylow | C37 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | C37⋊C3 |
| commutator, hypocenter | C37 | C3×C9 |
| Frattini | C3 | C3×C37⋊C3 |
| Fitting | C9×C37 | C3 |
| socle, cosocle | C3×C37 | C32 |
| prime radical | C3×C37⋊C3 | C3 |
| LCS | G ▷ C37 | |
| UCS | C1 ◁ C9 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 333 |
135 conjugacy classes — view the full character table.
| maximal subgroup of | C9×D37⋊C3, D9×C37⋊C3, (C37⋊C3)⋊D9, … |
| maximal normal subgroup of | C9×D37⋊C3, D9×C37⋊C3, (C37⋊C3)⋊D9, … |
| maximal quotient of | C2×C9×C37⋊C3, C9×#66524, #15688502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(999, 4);