Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C33 |
No aliases for this group.
| Sylow | C33 ⋊ #400 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×C3⋊(C3⋊D3) |
| commutator, socle, prime radical, cosocle | C23×C33 | C23 |
| Fitting | C33×#95 | C2 |
| hypercenter | #95 | C3⋊(C3⋊D3) |
| hypocenter | C33 | #400 |
| LCS | G ▷ C23×C33 ▷ C33 | |
| UCS | C1 ◁ C23 ◁ #95 | |
1876 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1876 conjugacy classes of subgroups — too many to draw. Only its 575 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 1728 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=e,c^2=f,b^a=bf,c^a=cef,g^a=g^2,h^a=h^2,i^a=i^2,c^b=cd282 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 29362);