Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C43 |
| direct product | C4×#88447 |
| derived, Sylow | H(3,𝔽3) ⋊ C43 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, hypocenter | H(3,𝔽3) | C43 |
| Frattini, socle | C23×C3 | C2×D32 |
| Fitting | C22×C4×H(3,𝔽3) | C22 |
| prime radical, cosocle | C23×H(3,𝔽3) | C23 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C22×C4 | |
1011 conjugacy classes of subgroups; only the 291 normal subgroups are shown.
This group has 1011 conjugacy classes of subgroups — too many to draw. Only its 291 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 6117);