Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #6249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
| direct product | C23×#6249 |
| Fitting | C23×H(3,𝔽3) ⋊ C8 |
| derived, Sylow | H(3,𝔽3) ⋊ C23×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #6249 |
| commutator, hypocenter | H(3,𝔽3) | C23×C8 |
| Frattini | C3 | C23×AGL(1,𝔽9) |
| Fitting | C23×H(3,𝔽3) | C8 |
| socle | C23×C3 | AGL(1,𝔽9) |
| prime radical | C23×#272 | C4 |
| cosocle | #1088 | C24 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23 | |
641 conjugacy classes of subgroups.
| transitive degree | μt(G) = 216 |
104 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#6249⋊C2 |
| maximal normal subgroup of | C23×#6249⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46138);