Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ #568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×#1113 |
| commutator, socle | C2×C33 | C23×C4 |
| Fitting | C33×M16 | C22 |
| prime radical | C2×#89042 | C2 |
| cosocle | C33×C4 | C24 |
| hypercenter | M16 | #1113 |
| hypocenter | C33 | #568 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ M16 | |
1256 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|d^2=e,e^2=f,b^a=bf,g^a=g^2,h^a=h^2,g^b=g^2,i^b=i^2,d^c=df144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 44526);