Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×A4⋊GL(2,𝔽3) |
| unambiguous semidirect product | A4⋊(C3×GL(2,𝔽3)) |
| Fitting | C22×C3×Q8 ⋊ C3⋊D3 |
| derived | A4×SL(2,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | A4⋊S4 |
| commutator, cosocle, hypocenter | A4×SL(2,𝔽3) | C2×C3 |
| Frattini | C2 | C3×A4⋊S4 |
| Fitting | C22×C3×Q8 | C3⋊D3 |
| socle | C23×C3 | C3⋊S4 |
| LCS | G ▷ A4×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
345 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
69 conjugacy classes — view the full character table.
| maximal subgroup of | C3×S4×GL(2,𝔽3), #72276552, D3×A4⋊GL(2,𝔽3) |
| maximal normal subgroup of | C3×S4×GL(2,𝔽3), #72276552, D3×A4⋊GL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46397);