Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C16 · Schur multiplier (H2) = C4 |
| direct product | C3×#10589102 |
| derived | C32 ⋊ C3×C4×C16 |
| Sylow | C33 ⋊ C4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C8 | D32 |
| commutator, hypocenter | C32 | C3×C4×C16 |
| Frattini | C2×C8 | C3×D32 |
| Fitting | C2×C33×C8 | C22 |
| socle, prime radical | C22×C33 | C2×C8 |
| cosocle | C2×C32×C8 | C22×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C2×C3×C8 | |
354 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
432 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11461);