Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of (C22⋊C27)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×(C22⋊C27)⋊C4 |
| unambiguous semidirect product | (C22⋊C27)⋊C42 |
| derived | C22⋊C27 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | (C22⋊C27)⋊C2 |
| commutator, hypocenter | C22⋊C27 | C42 |
| Frattini | C22×C9 | C2×S4 |
| Fitting | C23×C4×C9 | D3 |
| socle, prime radical | C24×C3 | C2×D9 |
| cosocle | C22×C22⋊C27 | C22 |
| LCS | G ▷ C22⋊C27 | |
| UCS | C1 ◁ C2×C4 | |
315 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2211);