Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×#11720108 |
| derived | Q8⋊C27 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | (C22⋊C27)⋊C2 |
| commutator, hypocenter | Q8⋊C27 | C2×C4 |
| Frattini | C22×C9 | C2×S4 |
| Fitting | C22×Q8×C9 | D3 |
| socle, prime radical | C23×C3 | (C22⋊C9)⋊C2 |
| cosocle | C2×Q8⋊C27 | C22 |
| LCS | G ▷ Q8⋊C27 | |
| UCS | C1 ◁ C23 | |
221 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 221 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1728 |
144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2219);