Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#5919 |
| Fitting | C22×C32×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C22×S4 |
| commutator, cosocle | C2×A4 | C23×C32 |
| Frattini | C2 | C22×C32×S4 |
| Fitting | C22×C32×Q8 | D3 |
| socle | C23×C32 | C22×D3 |
| hypercenter | C32×Q8 | S4 |
| hypocenter | A4 | C32×#41 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C32 ◁ C32×Q8 | |
1074 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
225 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47424);