Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32 |
| direct product | #41×C22⋊H(3,𝔽3) |
| Fitting | C22×C32×#41 ⋊ C3 |
| Sylow | C22×#41 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C22×C3×A4 |
| commutator, socle, cosocle | C23×C3 | C23×C32 |
| Frattini | C2×C3 | C23×C3×A4 |
| Fitting | C22×C32×#41 | C3 |
| hypercenter | C32×#41 | A4 |
| hypocenter | C22 | #41×H(3,𝔽3) |
| LCS | G ▷ C23×C3 ▷ C22 | |
| UCS | C1 ◁ C3×C4 ◁ C32×#41 | |
1250 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 1250 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30919);