Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C33×Q8 ⋊ C23 |
| Sylow | C33 ⋊ C22×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D33 |
| commutator, socle, cosocle | C2×C33 | C25 |
| Fitting | C33×Q8 | C23 |
| hypercenter | Q8 | D33 |
| hypocenter | C33 | C22×#41 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
4092 conjugacy classes of subgroups; only the 652 normal subgroups are shown.
This group has 4092 conjugacy classes of subgroups — too many to draw. Only its 652 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=f,e^2=f,e^a=ef,h^a=h^2,e^b=ef,g^b=g^2,e^c=ef,i^c=i^2,e^d=ef135 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47367);