Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #63419⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#63419⋊C2 |
| derived | C42⋊C9 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #63419⋊C2 |
| commutator, cosocle, hypocenter | C42⋊C9 | C22×C3 |
| Frattini | C22×C3 | C2×C3×S4 |
| Fitting | C32×#127 | D3 |
| socle | C22×C32 | C2×S4 |
| LCS | G ▷ C42⋊C9 | |
| UCS | C1 ◁ C3 | |
330 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
72 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12895);