Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #89101 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C22×#89101 |
| Fitting, socle | C24×C32 ⋊ C2×D3 |
| derived | C32×A4 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #89101 |
| commutator, cosocle, hypocenter | C32×A4 | C24 |
| Frattini | C1 | G |
| Fitting, socle | C24×C32 | C2×D3 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C22 | |
3130 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 3130 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
96 conjugacy classes — view the full character table.
| maximal subgroup of | C22×((C3⋊S4)⋊C3)⋊D3, A4×#89101, #72222977, … |
| maximal normal subgroup of | A4×#89101, #72222977, C22×D32×S4, … |
| maximal quotient of | D4×#89101 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47896);