Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C23×C33 ⋊ C23 |
| Sylow | C33 ⋊ C2×#120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D33 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Fitting | C23×C33 | C23 |
| hypercenter | C23 | D33 |
| hypocenter | C33 | C2×#120 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C23 | |
3198 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|d^a=df,h^a=h^2,d^b=df,g^b=g^2,d^c=de,i^c=i^2114 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#10595278 |
| maximal normal subgroup of | D3×#10595278 |
| maximal quotient of | #72220389, #72220954 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 39000);