Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C3×#10588351 |
| Fitting | C3×C9×#95 ⋊ C2 |
| Sylow | C3×C9 ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C3 | C22×D9 |
| commutator, cosocle | C23×C9 | C23×C3 |
| Frattini | C23×C3 | C22×C3×D3 |
| Fitting | C3×C9×#95 | C2 |
| socle | C23×C32 | C22×D3 |
| hypercenter | C3×#95 | D9 |
| hypocenter | C9 | C3×#392 |
| LCS | G ▷ C23×C9 ▷ C9 | |
| UCS | C1 ◁ C23×C3 ◁ C3×#95 | |
1042 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1042 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
306 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7379);