Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C3⋊((C22⋊C9)⋊C2) |
| commutator, cosocle, hypocenter | C3×Q8⋊C9 | C23 |
| Frattini | C2×C3 | C22×C3⋊S4 |
| Fitting | C32×2−1+4 | D3 |
| socle | C2×C32 | C22×S4 |
| hypercenter | D4 | C3⋊((C22⋊C9)⋊C2) |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ D4 | |
768 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 768 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 864 |
Polycyclic presentation:
2^3,3^3,2^3|b^2=i,d^3=f,g^2=i,h^2=i,c^a=ci,d^a=d^2f^2,e^a=e^2,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,g^d=h,h^d=ghi,h^g=hi99 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30568);