Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of C3⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C215×C3 |
| direct product | C25×C3×C3⋊D3 |
| Fitting, socle | C25×C33 ⋊ C2 |
| derived | C32 ⋊ C26×C3 |
| Sylow | C33 ⋊ C26 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25×C3 | C3⋊D3 |
| commutator, cosocle, hypocenter | C32 | C26×C3 |
| Frattini | C1 | G |
| Fitting, socle | C25×C33 | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C25×C3 | |
36892 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
576 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C24×C3⋊D3×AGL(1,𝔽7), C2×C3×C3⋊D3×#120 |
| maximal normal subgroup of | C2×C3×C3⋊D3×#120 |
| maximal quotient of | C24×C3⋊D3×AGL(1,𝔽7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47934);