Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C33 |
| unambiguous semidirect product | C3⋊(C3⋊#4621) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C3⋊#857) |
| commutator, prime radical, cosocle, hypocenter | C32×#95⋊C3 | C2 |
| Frattini | C23 | C3⋊(C3⋊S4) |
| Fitting | C32×#95 | D3 |
| socle | C23×C32 | S4 |
| LCS | G ▷ C32×#95⋊C3 | |
| UCS | C1 ◁ C2 | |
350 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 53 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|a^2=gh,e^2=g,f^2=ghi,b^a=b^2,c^a=c^2,d^a=d^2,e^a=efi,f^a=fghi,g^a=gi,h^a=hi,e^b=fh,f^b=efgh,g^b=ghi,h^b=i,i^b=hi,f^e=fgh63 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 23078);