Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#11722082 |
| Sylow | C33 ⋊ C2×#95 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | #1113 |
| commutator | C2×C33 | C2×C42 |
| Frattini | C23 | C2×#1113 |
| Fitting, socle, prime radical | C24×C33 | C22 |
| cosocle | C23×C33 | C23 |
| hypocenter | C33 | C2×#95 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C24 | |
2162 conjugacy classes of subgroups; only the 607 normal subgroups are shown.
This group has 2162 conjugacy classes of subgroups — too many to draw. Only its 607 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 28110);