Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #157⋊(#41⋊C3) |
| unambiguous semidirect product | #10596525⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×GO+(4,𝔽2) |
| commutator, cosocle | Q8×C3⋊D3 | C22×C3 |
| Fitting, hypocenter | C32×Q8 | C3×D4 |
| socle | C2×C32 | D4×A4 |
| LCS | G ▷ Q8×C3⋊D3 ▷ C32×Q8 | |
| UCS | C1 ◁ C2 | |
292 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^2,3,2^3,3^2,2|a^2=i,b^2=i,e^2=i,f^2=i,b^a=bd,g^a=g^2,h^a=g^2h,g^b=g^2h^2,e^c=ef,f^c=e,g^d=g^2,h^d=h^2,f^e=fi63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46284);