Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×C32×GL(2,ℤ/4) |
| Fitting | C25×C32 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C32 | C2×S4 |
| commutator, cosocle | C2×A4 | C23×C32 |
| Frattini | C2 | C22×C32×S4 |
| Fitting | C25×C32 | D3 |
| socle | C24×C32 | C2×D3 |
| hypercenter | C23×C32 | S4 |
| hypocenter | A4 | C2×C32×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22×C32 ◁ C23×C32 | |
1882 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1882 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
252 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C32×A4⋊S4 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47434);