Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Fitting, derived | C22×C32×Q8 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C3⋊D3)⋊(C22⋊A4) |
| commutator, Fitting, cosocle, hypocenter | C22×C32×Q8 | C2×C3 |
| Frattini | C2×C3 | D3×C22⋊A4 |
| socle | C23×C3 | D3×A4 |
| LCS | G ▷ C22×C32×Q8 | |
| UCS | C1 ◁ C2 | |
968 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 968 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2^4,3^2,2|e^2=i,f^2=i,g^a=g^2,h^a=h^2,c^b=cd,d^b=c,e^b=ef,f^b=e,g^b=gh,f^e=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30721);