Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of ((C3⋊S4)⋊C3)⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×((C3⋊S4)⋊C3)⋊C2 |
| unambiguous semidirect product | ((C3⋊S4)⋊C3)⋊(C2×C3) |
| Fitting | C22×C33 ⋊ C2×D3 |
| derived | C22⋊H(3,𝔽3) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | ((C3⋊S4)⋊C3)⋊C2 |
| commutator, cosocle, hypocenter | C22⋊H(3,𝔽3) | C22×C3 |
| Frattini | C3 | C3×D3×S4 |
| Fitting | C22×C33 | C2×D3 |
| socle | C22×C32 | D32 |
| LCS | G ▷ C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
332 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2923);