Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ #577 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D32×#67 |
| commutator, cosocle | C33×C4 | C24 |
| Frattini | C4 | C2×D33 |
| Fitting | C33×Q8 | C23 |
| socle | C2×C33 | C22×D4 |
| hypercenter | Q8 | D33 |
| hypocenter | C33 | #577 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
1232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|d^2=f,e^2=f,b^a=bf,h^a=h^2,d^b=df,g^b=g^2,d^c=de,e^c=ef,i^c=i^2,e^d=ef96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 37745);