Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #857 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | C2×C32×#857 |
| Fitting | C2×C32×C42 ⋊ D3 |
| derived | C42⋊C3 ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | #857 |
| commutator, cosocle, hypocenter | C42⋊C3 | C22×C32 |
| Frattini | C22 | C2×C32×S4 |
| Fitting | C2×C32×C42 | D3 |
| socle | C23×C32 | S4 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C2×C32 | |
552 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39002);