Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C33 ⋊ #336 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22×C4 | #1113 |
| commutator | C2×C33 | C4×C8 |
| Fitting, cosocle | C22×C33×C4 | C22 |
| socle, prime radical | C23×C33 | C2×C4 |
| hypocenter | C33 | #336 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22×C4 | |
786 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|a^2=e,b^2=c,c^2=f,b^a=bd,g^a=g^2,i^a=i^2,g^b=g^2,h^b=h^2216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11794);