Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | (C3⋊#4621)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | (C3⋊#857)⋊C3 |
| commutator, cosocle, hypocenter | C3×#95⋊C3 | C2×C3 |
| Frattini, socle | C23×C3 | C3×S4 |
| Fitting | C32×#95 | D3 |
| prime radical | #95⋊H(3,𝔽3) | C2 |
| LCS | G ▷ C3×#95⋊C3 | |
| UCS | C1 ◁ C2 | |
161 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 161 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3^3,2^5|a^2=gh,e^2=g,f^2=ghi,c^a=c^2,d^a=d^2,e^a=efi,f^a=fghi,g^a=gi,h^a=hi,c^b=cd^2,e^c=fh,f^c=efgh,g^c=ghi,h^c=i,i^c=hi,f^e=fgh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2814);