Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C33×#41 ⋊ C22 |
| Sylow | C33 ⋊ C2×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #89049 |
| commutator, cosocle | C33×C4 | C24 |
| Frattini | C4 | C22×#1113 |
| Fitting | C33×#41 | C22 |
| socle | C2×C33 | C22×D4 |
| hypercenter | #41 | #1113 |
| hypocenter | C33 | C2×#135 |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
1392 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|c^2=f,e^2=f,g^a=g^2,h^a=h^2,d^b=de,e^b=ef,g^b=g^2,i^b=i^2,e^d=ef126 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 44836);