Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C3 |
| direct product | C3×C3⋊(#504⋊C3) |
| unambiguous semidirect product | C3⋊(C3×#504⋊C3) |
| unambiguous semidirect product | #5140⋊C32 |
| unambiguous semidirect product | (C3×#5140)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | D3×A4 |
| commutator, hypocenter | C3×Q8 | C32×C8 |
| Frattini | C8 | C3×D3×A4 |
| Fitting | C32×#131 | C2×C3 |
| socle | C2×C32 | C8×A4 |
| prime radical | C32×#41⋊C3 | C4 |
| cosocle | C3×#131 | C2×C32 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C3×C8 | |
216 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
252 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23097);