Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C33 |
No aliases for this group.
| Fitting | C33×C32 ⋊ C2 |
| Sylow | C33 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11722169 |
| commutator, cosocle | C33×C16 | C22 |
| Frattini | C16 | C2×C3⋊(C3⋊D3) |
| Fitting | C33×C32 | C2 |
| socle | C2×C33 | D16 |
| hypercenter | C32 | C3⋊(C3⋊D3) |
| hypocenter | C33 | D32 |
| LCS | G ▷ C33×C16 ▷ C33×C8 ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
476 conjugacy classes of subgroups.
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^6,3^3|b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=h^2,i^a=i^2435 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12058);