Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C2×C33 |
| direct product | C3×(#41⋊C3)⋊A4 |
| unambiguous semidirect product | (#41⋊C3)⋊(C3×A4) |
| unambiguous semidirect product | #5945⋊C32 |
| unambiguous semidirect product | #5950⋊C32 |
| Fitting | C3×#585 ⋊ C32 |
| Sylow | #585 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C4 | A42 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C33 |
| Frattini | C2 | C2×C3×A42 |
| Fitting | C3×#585 | C32 |
| socle | C2×C3 | C2×A42 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C3×C4 | |
468 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
150 conjugacy classes — view the full character table.
| maximal subgroup of | #72242713, C3×(2−1+4⋊C3)⋊A4, C3×#11923520, … |
| maximal normal subgroup of | #72242713, C3×(2−1+4⋊C3)⋊A4, C3×#11923520, … |
| maximal quotient of | #72242713, #72245011 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46984);