Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #10595441⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D32×A4 |
| commutator, hypocenter | C32×Q8 | C2×C3×C4 |
| Fitting, cosocle | C32×#41 | C22×C3 |
| socle | C2×C32 | C2×C4×A4 |
| prime radical | C3×D3×#41⋊C3 | C2 |
| LCS | G ▷ C32×Q8 | |
| UCS | C1 ◁ C4 | |
364 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 58 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|b^2=d,d^2=i,e^2=i,f^2=i,b^a=bi,g^a=g^2,h^b=h^2,e^c=ef,f^c=e,f^e=fi108 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32218);