Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of S4×(C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| direct product | S4×(C3⋊D3)⋊C3 |
| unambiguous semidirect product | (C3⋊D3×S4)⋊C3 |
| Fitting | C22×H(3,𝔽3) ⋊ C2×D3 |
| derived | C32×A4 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×A4 | C22×C3 |
| Frattini | C3 | C3×D3×S4 |
| Fitting | C22×H(3,𝔽3) | C2×D3 |
| socle | C22×C3 | C3×D32 |
| LCS | G ▷ C32×A4 | |
| UCS | C1 | |
297 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 297 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.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
50 conjugacy classes — view the full character table.
| maximal subgroup of | S4×PSU(3,𝔽2)⋊C3, S4×((C3⋊D3)⋊C3)⋊C2, S4×(C3⋊D3)⋊A4, … |
| maximal normal subgroup of | S4×((C3⋊D3)⋊C3)⋊C2, C2×S4×(C3⋊D3)⋊C3 |
| maximal quotient of | #72222924, S4×(C3⋊D3)⋊A4, #72222965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2919);