Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊(C3⋊(C3⋊S4)) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C36 |
| direct product | C2×C3⋊(C3⋊(C3⋊S4)) |
| unambiguous semidirect product | (C2×C3)⋊(C3⋊(C3⋊S4)) |
| unambiguous semidirect product | (C2×A4)⋊(C3⋊(C3⋊D3)) |
| Fitting, socle | C23×C33 ⋊ D3 |
| derived | C33×A4 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊(C3⋊(C3⋊S4)) |
| commutator, cosocle, hypocenter | C33×A4 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C23×C33 | D3 |
| LCS | G ▷ C33×A4 | |
| UCS | C1 ◁ C2 | |
2484 conjugacy classes of subgroups; only the 483 normal subgroups are shown.
This group has 2484 conjugacy classes of subgroups — too many to draw. Only its 483 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 162 |
114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3589);