Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ #568 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×#1113 |
| commutator, socle | C2×C33 | C23×C4 |
| Fitting | C33×#41 | C22 |
| prime radical | C3×#41×C3⋊D3 | C2 |
| cosocle | C33×C4 | C24 |
| hypercenter | #41 | #1113 |
| hypocenter | C33 | #568 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ #41 | |
1360 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|b^2=e,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=df144 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 44831);