Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #1113 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C8 |
No aliases for this group.
| derived, Sylow | C33 ⋊ C82 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C42 | #1113 |
| commutator, hypocenter | C33 | C82 |
| Fitting, cosocle | C33×C42 | C22 |
| socle, prime radical | C22×C33 | C42 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C42 | |
458 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|a^2=d,b^2=c,c^2=e,d^2=f,g^a=g^2,i^a=i^2,g^b=g^2,h^b=h^2240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 11761);