Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #5450⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C22×C33 |
| Borel | B(3,𝔽4) |
| direct product | C3×#5450⋊C3 |
| unambiguous semidirect product | #5450⋊C32 |
| unambiguous semidirect product | #5464⋊C32 |
| unambiguous semidirect product | #5465⋊C32 |
| Fitting | C3×H(3,𝔽4) ⋊ C32 |
| derived, Sylow | H(3,𝔽4) ⋊ C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #5450⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽4) | C33 |
| Frattini | C22 | C3×A42 |
| Fitting | C3×H(3,𝔽4) | C32 |
| socle | C22×C3 | A42 |
| LCS | G ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C3 | |
298 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
57 conjugacy classes — view the full character table.
| derived subgroup of | #72241554, D3×#11921467 |
| maximal subgroup of | C3×AGL(2,𝔽4), C3×#11921467, #72241554, … |
| maximal normal subgroup of | C3×#11921467, #72241554, D3×#5450⋊C3, … |
| maximal quotient of | C2×C3×#5450⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 32237);