Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C27×D16 ⋊ C2 |
| Sylow | C27 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11720067 |
| commutator, cosocle | C16×C27 | C22 |
| Frattini | C9×C16 | C2×D3 |
| Fitting | C27×D16 | C2 |
| socle | C2×C3 | #63385 |
| hypercenter | D16 | D27 |
| hypocenter | C27 | D32 |
| LCS | G ▷ C16×C27 ▷ C8×C27 ▷ C4×C27 ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D16 | |
68 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^6,3^3|c^2=de,d^2=ef,e^2=f,g^3=h,h^3=i,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^2,c^b=cd,d^b=de,e^b=ef162 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 78);