Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C33 |
No aliases for this group.
| Fitting | C2×C33×C22⋊C4 ⋊ C2 |
| Sylow | C33 ⋊ #534 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C2×C33×C22⋊C4 | C2 |
| hypercenter | C2×C22⋊C4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #534 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 | |
4844 conjugacy classes of subgroups; only the 891 normal subgroups are shown.
This group has 4844 conjugacy classes of subgroups — too many to draw. Only its 891 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
Polycyclic presentation:
2^6,3^3|d^2=e,c^a=cf,d^a=de,g^a=g^2,h^a=h^2,i^a=i^2,d^b=df282 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 45680);