Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×((C22⋊C9)⋊C2)⋊C3 |
| commutator, cosocle, hypocenter | Q8⋊C9 | C23×C3 |
| Frattini, socle | C2×C3 | C22×C3×S4 |
| Fitting | C32×H(5,𝔽2) | D3 |
| hypercenter | Q8 | ((C22⋊C9)⋊C2)⋊C3 |
| LCS | G ▷ Q8⋊C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
525 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^3,3^3,2^3|b^2=i,c^2=i,e^3=f,g^2=i,h^2=i,e^a=e^2f^2,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,e^d=ef,g^e=h,h^e=ghi,h^g=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | C3×#13164432, #72253013 |
| maximal normal subgroup of | C3×#13164432, #72253013 |
| maximal quotient of | C3×#13164432 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30495);