Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C33 ⋊ #342 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #88814 |
| commutator | C22×C33 | C42 |
| Frattini, hypercenter | C24 | #1113 |
| Fitting, prime radical, cosocle | C24×C33 | C22 |
| socle | C22×C33 | C22⋊C4 |
| hypocenter | C33 | #342 |
| LCS | G ▷ C22×C33 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 ◁ C24 | |
1098 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^3|a^2=e,b^2=c,b^a=bd,c^a=cf,g^a=g^2,i^a=i^2,d^b=df,g^b=g^2,h^b=h^2162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11808);