Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C33 ⋊ C4×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C23 | D33 |
| commutator | C2×C33 | C2×C42 |
| Fitting, socle, cosocle | C23×C33 | C23 |
| prime radical | C23×C3⋊(C3⋊D3) | C22 |
| hypocenter | C33 | C4×C22⋊C4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C23 | |
1666 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=e,c^2=ef,c^a=cd,i^a=i^2,c^b=cd,g^b=g^2,h^c=h^2168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 22765);