Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #504 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×C3⋊#157 |
| commutator, socle | C2×C33 | C22×C8 |
| Fitting, prime radical | C33×#41 | C4 |
| cosocle | C3×C3⋊(C3⋊C8) | C23 |
| hypercenter | #41 | C3⋊#157 |
| hypocenter | C33 | #504 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ #41 | |
372 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 101 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|a^2=d,d^2=e,e^2=f,g^a=g^2,h^a=hi,i^a=hi^2,c^b=cf,h^d=h^2,i^d=i^2120 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39498);