Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #436 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D33 |
| commutator, socle | C22×C33 | C22×C4 |
| Fitting, cosocle | C2×C33×C4 | C23 |
| prime radical | C2×#6295 | C4 |
| hypocenter | C33 | #436 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
690 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 690 conjugacy classes of subgroups — too many to draw. Only its 116 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=e,b^2=e,c^2=de,e^2=f,b^a=bd,c^a=cdf,i^a=i^2,c^b=cdf,g^b=g^2,h^c=h^2144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 21669);