Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C3⋊D3×S4 |
| commutator, cosocle | C2×C32×SL(2,𝔽3) | C22 |
| Fitting | C2×C32×Q8 | C2×D3 |
| socle | C22×C32 | C2×S4 |
| hypocenter | C32×SL(2,𝔽3) | D4 |
| LCS | G ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
738 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3^3,2^3|b^2=c,g^2=i,h^2=i,b^a=bc,e^a=e^2,f^a=f^2,d^b=d^2,g^b=gh,h^b=hi,g^d=h,h^d=ghi,h^g=hi87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32031);