Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×#11724187 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C3⋊D3×S4 |
| commutator, cosocle | C2×C32×A4 | C23 |
| Frattini | C2 | C2×C3⋊D3×S4 |
| Fitting, socle | C24×C32 | C2×D3 |
| hypocenter | C32×A4 | C2×D4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C22 | |
1654 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 1654 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46842);