Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #10595351⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D32×A4 |
| commutator, Fitting, cosocle | C2×C32×Q8 | C22×C3 |
| socle | C22×C32 | C22×A4 |
| prime radical | C2×C32×SL(2,𝔽3) | C22 |
| hypocenter | C32×Q8 | C3×Q8 |
| LCS | G ▷ C2×C32×Q8 ▷ C32×Q8 | |
| UCS | C1 ◁ C22 | |
348 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^2,3,2^3,3^2,2|a^2=d,b^2=d,e^2=i,f^2=i,b^a=bdi,g^a=g^2,h^b=h^2,e^c=ef,f^c=e,f^e=fi105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32210);