Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C22×C3×Dic32 |
| derived | C32 ⋊ C22×C3×C42 |
| Sylow | C33 ⋊ C22×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24×C3 | D32 |
| commutator, hypocenter | C32 | C22×C3×C42 |
| Frattini | C22 | C22×C3×D32 |
| Fitting, socle, prime radical | C24×C33 | C22 |
| cosocle | C22×C32 | C24×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C24×C3 | |
2834 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
432 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 41134);