Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ #536 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D33 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C2×C33×C4 | C23 |
| prime radical | C2×#89042 | C2 |
| hypercenter | C2×C4 | D33 |
| hypocenter | C33 | #536 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
1422 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 1422 conjugacy classes of subgroups — too many to draw. Only its 196 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=f,b^2=e,c^2=ef,d^2=ef,c^a=cf,d^a=def,h^a=h^2,c^b=ce,d^b=def,g^b=g^2,d^c=def,i^c=i^2108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 36540);