Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C42×D27 |
| Fitting | C2×C42×C27 ⋊ C2 |
| derived, Sylow | C27 ⋊ C22×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C42 | D27 |
| commutator, hypocenter | C27 | C22×C42 |
| Frattini | C22×C9 | C23×D3 |
| Fitting | C2×C42×C27 | C2 |
| socle | C23×C3 | C22×D9 |
| prime radical | C23×D27 | C22 |
| cosocle | C22×C27 | C24 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C2×C42 | |
996 conjugacy classes of subgroups; only the 411 normal subgroups are shown.
This group has 996 conjugacy classes of subgroups — too many to draw. Only its 411 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 864 |
480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2272);