Finite Group DatabasebetaFinite Group Databasebeta| order | 3645 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C5×C9 · Schur multiplier (H2) = C33 |
| direct product | C5×#10624704 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C9 | C3×H(3,𝔽3) |
| commutator | C32 | C32×C5×C9 |
| Frattini, cosocle | C3×C9 | C33×C5 |
| socle | C3×C5 | C3×C32⋊C9 |
| prime radical | C3×C5×H(3,𝔽3) | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C5×C9 ◁ C32×C5×C9 ◁ G | |
312 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 86 |
| transitive degree | μt(G) = 405 |
525 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3645, 264);