Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #11724205 |
| commutator, cosocle | C32×C4×A4 | C22 |
| Frattini, hypercenter | C4 | D3×C3⋊S4 |
| Fitting | C22×C32×C4 | C2×D3 |
| socle | C23×C32 | #67 |
| hypocenter | C32×A4 | D8 |
| LCS | G ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
588 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 60 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:
2^4,3^3,2^2|a^2=c,c^2=d,b^a=bc,e^a=e^2,f^a=f^2,h^a=hi,c^b=cd,g^b=g^2,h^e=i,i^e=hi75 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32071);