Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D33 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C3⋊C8×#624 |
| derived, Sylow | C33 ⋊ C2×C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D33 |
| commutator, hypocenter | C33 | C2×C4×C8 |
| Fitting, cosocle | C2×C33×C4 | C23 |
| socle | C22×C33 | C22×C4 |
| prime radical | C22×C3×C3⋊D3 | C2×C4 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
834 conjugacy classes of subgroups.
| 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, 21329);