Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting | C33×C42 ⋊ C4 |
| Sylow | C33 ⋊ #383 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×C3⋊#157 |
| commutator, socle | C22×C33 | C22×C4 |
| Fitting | C33×C42 | C4 |
| prime radical, cosocle | C22×C3×C3⋊D3 | C23 |
| hypercenter | C42 | C3⋊#157 |
| hypocenter | C33 | #383 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C42 | |
552 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|a^2=d,b^2=e,c^2=f,b^a=bf,c^a=cef,g^a=g^2,h^a=hi,i^a=hi^2,h^d=h^2,i^d=i^2156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39397);