Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×D3×C3⋊GL(2,𝔽3) |
| Fitting | C2×C32×Q8 ⋊ C2×D3 |
| derived | C32×SL(2,𝔽3) ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×C3⋊S4 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C23 |
| Frattini | C2 | C2×D3×C3⋊S4 |
| Fitting | C2×C32×Q8 | C2×D3 |
| socle | C22×C32 | C2×S4 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
1361 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 1361 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
90 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D3×C3⋊AGL(2,𝔽3), C2×D3×SO+(4,𝔽3), D3×#10596446 |
| maximal normal subgroup of | D3×#10596446 |
| maximal quotient of | C2×D3×C3⋊AGL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46875);