Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#11720651 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | (C22⋊C27)⋊C2 |
| commutator, cosocle, hypocenter | Q8⋊C27 | C23 |
| Frattini | C2×C9 | C22×S4 |
| Fitting | C2×C9×#41 | D3 |
| socle | C22×C3 | C2×(C22⋊C9)⋊C2 |
| LCS | G ▷ Q8⋊C27 | |
| UCS | C1 ◁ C2×C4 | |
373 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 864 |
144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11366);