Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C3×C4 |
| direct product | C4×C3⋊(C9⋊C16) |
| derived, Sylow | C3×C9 ⋊ C4×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C8 | C3⋊D9 |
| commutator, hypocenter | C3×C9 | C4×C16 |
| Frattini | C2×C3×C8 | C2×C3⋊D3 |
| Fitting | C3×C4×C8×C9 | C2 |
| socle, prime radical | C22×C32 | C2×C3⋊C8 |
| cosocle | C2×C3×C8×C9 | C22 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 ◁ C4×C8 | |
290 conjugacy classes of subgroups; only the 227 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 227 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1728 |
480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1802);