Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #11720758 |
| commutator, Fitting, cosocle, hypocenter | C32×#95 | C2×C3 |
| Frattini, socle | C23×C3 | D3×A4 |
| prime radical | #95⋊H(3,𝔽3) | C2 |
| LCS | G ▷ C32×#95 | |
| UCS | C1 ◁ C2 | |
168 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2,3,2^2,3^2,2^3|a^2=gh,c^2=g,d^2=gi,c^a=chi,d^a=di,e^a=e^2,f^a=f^2,c^b=cd,d^b=chi,e^b=ef,g^b=ghi,h^b=i,i^b=hi,d^c=dgh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2893);