Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32⋊S4 |
| commutator, cosocle | A4×C3⋊Dic3 | C22 |
| Fitting, socle | C23×C32 | #67 |
| prime radical | C2×C32×A4 | D4 |
| hypocenter | C32×A4 | Q16 |
| LCS | G ▷ A4×C3⋊Dic3 ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 | |
252 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^4,3^3,2^2|a^2=d,b^2=d,c^2=d,b^a=bcd,c^a=cd,e^a=e^2,f^a=f^2,g^a=f^2g,h^a=hi,c^b=cd,f^b=f^2g^2,f^c=f^2,g^c=g^2,h^e=i,i^e=hi45 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46281);