Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊(GL(2,𝔽3)⋊A4) |
| Fitting | C3×H(5,𝔽2) ⋊ C3×D3 |
| derived | C3×#994 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | A4×C3⋊S4 |
| commutator, cosocle, hypocenter | C3×#994 | C2×C3 |
| Fitting | C3×H(5,𝔽2) | C3×D3 |
| socle | C2×C3 | A4×S4 |
| LCS | G ▷ C3×#994 | |
| UCS | C1 ◁ C2 | |
296 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2,3,2^2,3^2,2^3|c^2=i,d^2=i,g^2=i,h^2=i,e^a=e^2,f^a=f^2,g^a=gh,h^a=hi,c^b=d,d^b=cdi,d^c=di,g^e=h,h^e=ghi,h^g=hi54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×GL(2,𝔽3)⋊A4, #72242411, #72242414, … |
| maximal normal subgroup of | D3×GL(2,𝔽3)⋊A4, #72242411, #72242414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46388);