Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C22×C3×D3×SL(2,𝔽3) |
| unambiguous semidirect product | C3⋊(C23×C3×SL(2,𝔽3)) |
| unambiguous semidirect product | (C2×C3)⋊(C22×C3×SL(2,𝔽3)) |
| Fitting | C22×C32×Q8 ⋊ C2×C3 |
| derived | C3×Q8 ⋊ C23×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C23×C32 |
| Frattini | C2 | C22×C3×D3×A4 |
| Fitting | C22×C32×Q8 | C2×C3 |
| socle | C23×C32 | C2×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C23×C3 | |
1639 conjugacy classes of subgroups; only the 292 normal subgroups are shown.
This group has 1639 conjugacy classes of subgroups — too many to draw. Only its 292 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, 47505);