Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C9⋊#1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24×C3 |
| direct product | C2×C9⋊#1020 |
| unambiguous semidirect product | (C2×C9)⋊#1020 |
| Fitting | C25×C9 ⋊ D3 |
| derived | C9×C22⋊A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C9⋊#1020 |
| commutator, cosocle, hypocenter | C9×C22⋊A4 | C22 |
| Frattini | C3 | C2×C3⋊#1020 |
| Fitting | C25×C9 | D3 |
| socle | C25×C3 | C3⋊D3 |
| LCS | G ▷ C9×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
822 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 822 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
84 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72229294, C2×C3×C9⋊#1020 |
| maximal normal subgroup of | C2×C3×C9⋊#1020 |
| maximal quotient of | C2×C3×C9⋊#1020 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46152);