Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of (C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C210×C3 |
| direct product | C24×(C3⋊Dic3)⋊C3 |
| derived | C32 ⋊ C24×C3×C4 |
| Sylow | H(3,𝔽3) ⋊ C24×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25 | (C3⋊D3)⋊C3 |
| commutator, hypocenter | C32 | C24×C3×C4 |
| Frattini | C2×C3 | C24×C3×D3 |
| Fitting, prime radical | C25×H(3,𝔽3) | C2 |
| socle | C25×C3 | C3×D3 |
| cosocle | C2×C32 | C25×C3 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C25 | |
5515 conjugacy classes of subgroups; only the 2484 normal subgroups are shown.
This group has 5515 conjugacy classes of subgroups — too many to draw. Only its 2484 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
320 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31030);