Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C33 · Schur multiplier (H2) = C23×C33 |
| direct product | C2×C32×2−1+4⋊C3 |
| unambiguous semidirect product | (C2×2−1+4)⋊C33 |
| unambiguous semidirect product | (C2×C3×2−1+4)⋊C32 |
| Fitting | C2×C32×2−1+4 ⋊ C3 |
| Sylow | C2×2−1+4 ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C23×C33 |
| Frattini | C2 | C23×C32×A4 |
| Fitting | C2×C32×2−1+4 | C3 |
| hypercenter | C2×C32×D4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C32 ◁ C2×C32×D4 | |
1912 conjugacy classes of subgroups; only the 562 normal subgroups are shown.
This group has 1912 conjugacy classes of subgroups — too many to draw. Only its 562 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
342 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47766);