Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of (C22⋊C9)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C16 · Schur multiplier (H2) = C2 |
| direct product | C3×(C22⋊C9)⋊C16 |
| derived | C22⋊C9 ⋊ C3×C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | (C22⋊C9)⋊C2 |
| commutator, hypocenter | C22⋊C9 | C3×C16 |
| Frattini | C3×C8 | C3×S4 |
| Fitting | C22×C32×C8 | D3 |
| socle, prime radical | C23×C32 | C3⋊C8 |
| cosocle | C8×C22⋊C9 | C2×C3 |
| LCS | G ▷ C22⋊C9 | |
| UCS | C1 ◁ C3×C8 | |
138 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 144 |
216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2828);