Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ #532 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#1113 |
| commutator, socle, cosocle | C22×C33 | C24 |
| Frattini | C22 | C22×#1113 |
| Fitting | C33×C42 | C22 |
| hypercenter | C42 | #1113 |
| hypocenter | C33 | #532 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C4 ◁ C42 | |
1566 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|c^2=f,d^2=e,b^a=bf,g^a=g^2,h^a=h^2,d^b=de,g^b=g^2,i^b=i^2180 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 44212);