Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C23×C32 |
No aliases for this group.
| Sylow | C26 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3 | #10596782 |
| commutator, Fitting, socle, cosocle | C26×C3 | C32 |
| hypocenter | C26 | H(3,𝔽3) |
| LCS | G ▷ C26×C3 ▷ C26 | |
| UCS | C1 ◁ C3 | |
922 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 922 conjugacy classes of subgroups — too many to draw. Only its 24 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
3^3,2^6|b^a=bc,d^a=de,e^a=d,h^a=hi,i^a=h,d^b=de,e^b=d,f^b=fg,g^b=f56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46135);