Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C33 ⋊ #449 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11724113 |
| commutator, cosocle | C3×C4×C3⋊D3 | C23 |
| Frattini, hypercenter | C4 | D3×GO+(4,𝔽2) |
| Fitting | C33×C4 | C2×D4 |
| socle | C2×C33 | #120 |
| hypocenter | C33 | #449 |
| LCS | G ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
626 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 626 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=e,d^2=f,e^2=f,b^a=bd,d^a=df,e^a=ef,h^a=h^2,i^a=h^2i,c^b=ce,h^b=h^2i^2,d^c=df,e^c=ef,g^c=g^2,h^d=h^2,i^d=i^263 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31270);