Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | D3×#63817 |
| Sylow | C33 ⋊ C2×#131 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D33 |
| commutator, socle | C2×C33 | C23×C4 |
| Fitting | C2×C33×C4 | C23 |
| prime radical | D3×#3828 | C2 |
| cosocle | C33×C4 | C24 |
| hypercenter | C2×C4 | D33 |
| hypocenter | C33 | C2×#131 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
1392 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 36550);