Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29×C3 |
No aliases for this group.
| Fitting | H(3,𝔽3)×2−1+4 ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×(C3⋊D3)⋊C3 |
| commutator, cosocle | C2×C32 | C25×C3 |
| Frattini, socle | C2×C3 | C24×C3×D3 |
| Fitting | H(3,𝔽3)×2−1+4 | C2 |
| hypercenter | 2−1+4 | (C3⋊D3)⋊C3 |
| hypocenter | C32 | C3×#585 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
2981 conjugacy classes of subgroups; only the 954 normal subgroups are shown.
This group has 2981 conjugacy classes of subgroups — too many to draw. Only its 954 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^5,3,2,3^2|c^2=g,e^2=g,e^a=eg,h^a=h^2,i^a=i^2,d^b=dg,e^c=eg,h^f=hi170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31023);