Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#11720144 |
| Fitting | C2×C4⋊C4×C27 ⋊ C2 |
| Sylow | C27 ⋊ C2×#121 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D27 |
| commutator, cosocle | C22×C27 | C24 |
| Frattini | C22×C9 | C23×D3 |
| Fitting | C2×C4⋊C4×C27 | C2 |
| socle | C23×C3 | C22×D9 |
| hypercenter | C2×C4⋊C4 | D27 |
| hypocenter | C27 | C2×#121 |
| LCS | G ▷ C22×C27 ▷ C27 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
852 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 852 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 864 |
288 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2308);