Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #4656 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C25×C3 |
| direct product | C32×#4671 |
| Fitting | C33×#95 ⋊ C2 |
| Sylow | C33 ⋊ #392 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C32 | C22×D3 |
| commutator, cosocle | C23×C3 | C23×C32 |
| Frattini | C23 | C22×C32×D3 |
| Fitting | C33×#95 | C2 |
| socle | C23×C33 | C23 |
| hypercenter | C32×#95 | D3 |
| hypocenter | C3 | C32×#392 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23×C32 ◁ C32×#95 | |
2236 conjugacy classes of subgroups; only the 414 normal subgroups are shown.
This group has 2236 conjugacy classes of subgroups — too many to draw. Only its 414 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
378 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 28172);