Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D9⋊(#95⋊C3) |
| Fitting, derived | C9×#95 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9⋊(C42⋊C3) |
| commutator, Fitting, cosocle, hypocenter | C9×#95 | C2×C3 |
| Frattini, socle | C23×C3 | D3×A4 |
| LCS | G ▷ C9×#95 | |
| UCS | C1 ◁ C2 | |
230 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2,3,2^2,3,2^3,3|c^2=f,d^2=fg,e^3=i^2,e^a=e^2i,i^a=i^2,c^b=dh,d^b=cd,e^b=ei,f^b=fg,g^b=h,h^b=gh,d^c=dfh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2866);