Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #89101 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| derived | C32×A4 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #89101 |
| commutator, hypocenter | C32×A4 | C2×C8 |
| Fitting | C22×C32×C4 | C2×D3 |
| socle | C23×C32 | C4×D3 |
| prime radical | C2×C3×C3⋊S4 | C4 |
| cosocle | C32×C4×A4 | C22 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 ◁ C4 | |
402 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 52 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|b^2=c,c^2=d,e^a=e^2,f^a=f^2,h^a=hi,f^b=f^2,g^b=g^2,h^e=i,i^e=hi96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32134);