Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #6196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C23×#6200 |
| Sylow | H(3,𝔽3) ⋊ C23×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle | C2×H(3,𝔽3) | C25 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C24×H(3,𝔽3) | C22 |
| socle | C24×C3 | D32 |
| hypocenter | H(3,𝔽3) | C23×D4 |
| LCS | G ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C24 | |
5515 conjugacy classes of subgroups; only the 795 normal subgroups are shown.
This group has 5515 conjugacy classes of subgroups — too many to draw. Only its 795 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30880);