Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C33×D8 ⋊ C22 |
| Sylow | C33 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×#1113 |
| commutator, cosocle | C33×C4 | C24 |
| Frattini | C4 | C22×#1113 |
| Fitting | C33×D8 | C22 |
| socle | C2×C33 | C22×D4 |
| hypercenter | D8 | #1113 |
| hypocenter | C33 | #575 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
1268 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|e^2=f,d^a=df,g^a=g^2,h^a=h^2,c^b=cf,g^b=g^2,i^b=i^2,d^c=de,e^c=ef,e^d=ef93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 44530);