Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | #89101 |
| commutator | C2×C32×A4 | C2×C4 |
| Fitting, socle | C24×C32 | C2×D3 |
| prime radical, cosocle | C22×C32×A4 | C22 |
| hypocenter | C32×A4 | C4⋊C4 |
| LCS | G ▷ C2×C32×A4 ▷ C32×A4 | |
| UCS | C1 ◁ C22 | |
586 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 586 conjugacy classes of subgroups — too many to draw. Only its 74 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=c,b^a=bc,e^a=e^2,f^a=f^2,h^a=hi,f^b=f^2,g^b=g^2,h^e=i,i^e=hi84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32176);