Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C24×((C3⋊D3)⋊C3)⋊C2 |
| unambiguous semidirect product | ((C3⋊D3)⋊C3)⋊C25 |
| unambiguous semidirect product | #272⋊C25 |
| Fitting | C24×H(3,𝔽3) ⋊ C22 |
| derived, Sylow | H(3,𝔽3) ⋊ C26 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | ((C3⋊D3)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C26 |
| Frattini | C3 | C24×D32 |
| Fitting | C24×H(3,𝔽3) | C22 |
| socle | C24×C3 | D32 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C24 | |
20523 conjugacy classes of subgroups; only the 3707 normal subgroups are shown.
This group has 20523 conjugacy classes of subgroups — too many to draw. Only its 3707 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
176 conjugacy classes — view the full character table.
| maximal subgroup of | C24×AGL(2,𝔽3), #120×((C3⋊D3)⋊C3)⋊C2 |
| maximal normal subgroup of | #120×((C3⋊D3)⋊C3)⋊C2 |
| maximal quotient of | H(5,𝔽2)×((C3⋊D3)⋊C3)⋊C2 |
| automorphism group of | #88448, #88450, #3750⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46165);