Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #1088 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C24×#1088 |
| Fitting | C24×H(3,𝔽3) ⋊ C4 |
| derived, Sylow | H(3,𝔽3) ⋊ C24×C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24×C3 | #157 |
| commutator, hypocenter | H(3,𝔽3) | C24×C4 |
| Frattini | C3 | C24×#157 |
| Fitting | C24×H(3,𝔽3) | C4 |
| prime radical | C24×#272 | C2 |
| cosocle | #272 | C25 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C24×C3 | |
3539 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46163);