Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #1088⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C23×#1088⋊C2 |
| unambiguous semidirect product | #1088⋊C24 |
| unambiguous semidirect product | (((C3⋊D3)⋊C3)⋊C2)⋊C24 |
| Fitting | C23×H(3,𝔽3) ⋊ D4 |
| Sylow | H(3,𝔽3) ⋊ C23×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #1088⋊C2 |
| commutator, cosocle | #272 | C25 |
| Frattini | C3 | C23×GO+(4,𝔽2) |
| Fitting | C23×H(3,𝔽3) | D4 |
| socle | C23×C3 | GO+(4,𝔽2) |
| hypocenter | H(3,𝔽3) | C23×D4 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C23 | |
3539 conjugacy classes of subgroups; only the 457 normal subgroups are shown.
This group has 3539 conjugacy classes of subgroups — too many to draw. Only its 457 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
104 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#6249⋊C2 |
| maximal normal subgroup of | C23×#6249⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46139);