Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #6295 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C4×#89042 |
| Sylow | C33 ⋊ C4×#41 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | #1113 |
| commutator | C2×C33 | C23×C4 |
| Frattini | C22 | C22×#1113 |
| Fitting | C33×C42 | C22 |
| socle, cosocle | C22×C33 | C24 |
| prime radical | C2×#89042 | C2 |
| hypocenter | C33 | C4×#41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C42 | |
1594 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 1594 conjugacy classes of subgroups — too many to draw. Only its 287 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, 44210);