Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | 3−1+2×H(5,𝔽2) ⋊ C2 |
| Sylow | 3−1+2 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×AGL(1,ℤ/9) |
| commutator, cosocle | C2×C9 | C25×C3 |
| Frattini, socle | C2×C3 | C24×C3×D3 |
| Fitting | 3−1+2×H(5,𝔽2) | C2 |
| hypercenter | H(5,𝔽2) | AGL(1,ℤ/9) |
| hypocenter | C9 | C3×#585 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
2513 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^5,3,2,3^2|c^2=g,h^3=i,e^a=eg,h^a=h^2i^2,i^a=i^2,d^b=dg,e^c=eg,h^f=hi^2170 conjugacy classes — view the full character table.
| maximal subgroup of | #72245909 |
| maximal normal subgroup of | #72245909 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31017);