Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | C9⋊#5931 |
| Fitting | C22×Q8×C9 ⋊ D3 |
| derived | C9×C22⋊SL(2,𝔽3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C9⋊#1020 |
| commutator, cosocle, hypocenter | C9×C22⋊SL(2,𝔽3) | C2 |
| Frattini | C2×C3 | C3⋊#1020 |
| Fitting | C22×Q8×C9 | D3 |
| socle | C23×C3 | C3⋊S4 |
| LCS | G ▷ C9×C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
287 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2,3^3,2^5|c^3=d,g^2=i,h^2=i,b^a=b^2,c^a=c^2d^2,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,e^b=f,f^b=ef,g^b=h,h^b=ghi,h^g=hi63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30604);