Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×#11722724 |
| Sylow | C33 ⋊ C2×#97 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3×#157 |
| commutator | C2×C33 | C2×C42 |
| Frattini | C22 | C2×D3×#157 |
| Fitting, socle, prime radical | C23×C33 | C2×C4 |
| cosocle | C2×C3×C3⋊Dic3 | C23 |
| hypocenter | C33 | C2×#97 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
616 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 616 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32644);