Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #1113 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C24×#1113 |
| Fitting, socle | C24×C33 ⋊ C22 |
| derived, Sylow | C33 ⋊ C26 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | #1113 |
| commutator, cosocle, hypocenter | C33 | C26 |
| Frattini | C1 | G |
| Fitting, socle | C24×C33 | C22 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C24 | |
24978 conjugacy classes of subgroups; only the 4215 normal subgroups are shown.
This group has 24978 conjugacy classes of subgroups — too many to draw. Only its 4215 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47930);