Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D33 |
| commutator, socle | C2×C33 | C22×C8 |
| Fitting, cosocle | C33×C8 | C23 |
| prime radical | #89020 | C4 |
| hypocenter | C33 | #504 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C8 | |
474 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 98 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|b^2=d,c^2=d,d^2=e,e^2=f,c^a=cf,i^a=i^2,g^b=g^2,h^c=h^2180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 21254);