Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×S4 |
| commutator, cosocle, hypocenter | C32×SL(2,𝔽3) | C23 |
| Fitting | C32×Q8 | C22×D3 |
| socle | C2×C32 | C22×S4 |
| LCS | G ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
709 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 709 conjugacy classes of subgroups — too many to draw. Only its 52 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:
2^3,3^3,2^3|g^2=i,h^2=i,b^a=bi,e^a=e^2,f^b=f^2,d^c=d^2,g^c=gh,h^c=hi,g^d=h,h^d=ghi,h^g=hi63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46311);