Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of D27 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C8×D27 |
| Fitting | C4×C8×C27 ⋊ C2 |
| derived, Sylow | C27 ⋊ C2×C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C8 | D27 |
| commutator, hypocenter | C27 | C2×C4×C8 |
| Frattini | C2×C4×C9 | C22×D3 |
| Fitting | C4×C8×C27 | C2 |
| socle | C22×C3 | C2×C4×D9 |
| prime radical | C22×D27 | C2×C4 |
| cosocle | C2×C4×C27 | C23 |
| LCS | G ▷ C27 | |
| UCS | C1 ◁ C4×C8 | |
324 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 147 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 864 |
480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 254);