Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C23×C33 |
| direct product | C32×#5950 |
| Fitting | C32×#585 ⋊ C3 |
| Sylow | #585 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C2×C33 |
| Frattini | C2 | C2×C32×C22⋊A4 |
| Fitting | C32×#585 | C3 |
| socle | C2×C32 | C2×C22⋊A4 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C32×C4 | |
1278 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1278 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
198 conjugacy classes — view the full character table.
| maximal subgroup of | #72242713, #72245011, C32×#87547, … |
| maximal normal subgroup of | #72242713, #72245011, C32×#87547, … |
| maximal quotient of | 3−1+2×#5950 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47772);