Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C33 ⋊ #346 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×AGL(1,𝔽9) |
| commutator | C2×C33 | C4×C8 |
| Fitting, socle, prime radical | C22×C33 | C2×C8 |
| cosocle | C2×C3×#622 | C22 |
| hypocenter | C33 | #346 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22 | |
154 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 154 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) = 192 |
Polycyclic presentation:
2^6,3^3|a^2=c,b^2=d,c^2=e,e^2=f,b^a=bf,h^a=h^2i^2,i^a=h^2,g^b=g^2,h^c=h^2i,i^c=hi,h^e=h^2,i^e=i^284 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31126);