Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D33 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×#88949 |
| derived, Sylow | C33 ⋊ C24×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D33 |
| commutator, hypocenter | C33 | C24×C4 |
| Frattini | C2 | C22×D33 |
| Fitting, socle | C23×C33 | C23 |
| prime radical | C23×#1113 | C2 |
| cosocle | C2×C33 | C25 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C23 | |
6210 conjugacy classes of subgroups; only the 1132 normal subgroups are shown.
This group has 6210 conjugacy classes of subgroups — too many to draw. Only its 1132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47382);