Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32 |
| unambiguous semidirect product | C4⋊(#41⋊H(3,𝔽3)) |
| unambiguous semidirect product | #550⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×C4⋊(#41⋊C3))⋊C3 |
| Fitting | C32×#550 ⋊ C3 |
| Sylow | #550 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22×C3 | C22×C3×A4 |
| commutator, cosocle | C2×C3×Q8 | C22×C32 |
| Fitting | C32×#550 | C3 |
| hypercenter | C32×C4⋊C4 | A4 |
| hypocenter | Q8 | D4×H(3,𝔽3) |
| LCS | G ▷ C2×C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C3 ◁ C32×C4⋊C4 | |
501 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 501 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^2,3^2,2,3,2^3|a^2=e,b^2=i,g^2=i,h^2=i,b^a=be,d^c=df,g^c=gh,h^c=g,h^g=hi155 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18261);