Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D9×S4 |
| commutator, hypocenter | C9×SL(2,𝔽3) | C2×C4 |
| Frattini | C3×C4 | D3×S4 |
| Fitting | C9×#41 | C2×D3 |
| socle | C2×C3 | Dic3×S4 |
| prime radical | C3×#985 | D3 |
| cosocle | C9×#41⋊C3 | C22 |
| LCS | G ▷ C9×SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
200 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3^3,2^3|a^2=c,c^2=i,e^3=f,g^2=i,h^2=i,b^a=bi,e^a=e^2f^2,f^a=f^2,d^b=d^2,g^b=gh,h^b=hi,g^d=h,h^d=ghi,h^g=hi90 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12300);