Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | D32×#41⋊C3 |
| unambiguous semidirect product | (D32×#41)⋊C3 |
| Fitting | C32×#41 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D32×A4 |
| commutator, cosocle, hypocenter | C32×Q8 | C23×C3 |
| Frattini | C2 | C2×D32×A4 |
| Fitting | C32×#41 | C22×C3 |
| socle | C2×C32 | C23×A4 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C4 | |
991 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 991 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46968);