Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #64188 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×C3×#64188 |
| derived | C22⋊(C22⋊C9) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #64188 |
| commutator, cosocle, hypocenter | C22⋊(C22⋊C9) | C22×C3 |
| Frattini | C3 | C2×C3×#1020 |
| Fitting, socle | C25×C32 | D3 |
| LCS | G ▷ C22⋊(C22⋊C9) | |
| UCS | C1 ◁ C2×C3 | |
1046 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 1046 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×C9⋊#1020, C2×C3×#11724027 |
| maximal quotient of | C2×#72229320 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46149);