Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C8 |
| direct product | C8×C27⋊C8 |
| unambiguous semidirect product | C27⋊C82 |
| derived, Sylow | C27 ⋊ C82 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C8 | D27 |
| commutator, hypocenter | C27 | C82 |
| Frattini | C42×C9 | C2×D3 |
| Fitting | C4×C8×C27 | C2 |
| socle, prime radical | C22×C3 | C4×Dic9 |
| cosocle | C42×C27 | C22 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C4×C8 | |
148 conjugacy classes of subgroups.
| transitive degree | μt(G) = 1728 |
480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12);