Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C32 |
| direct product | C2×C3⋊#6258 |
| unambiguous semidirect product | (C2×C3)⋊#6258 |
| Fitting | C23×C33 ⋊ D3 |
| derived | C3×C22⋊H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | C3⋊(C3⋊S4) |
| commutator, cosocle, hypocenter | C3×C22⋊H(3,𝔽3) | C22 |
| Frattini | C3 | C2×C3⋊(C3⋊S4) |
| Fitting | C23×C33 | D3 |
| socle | C23×C32 | C3⋊D3 |
| LCS | G ▷ C3×C22⋊H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
668 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 668 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
78 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C3⋊#6258, C2×((C3⋊S4)⋊C3)⋊D3, C2×D3×#6258, … |
| maximal normal subgroup of | C22×C3⋊#6258, C2×((C3⋊S4)⋊C3)⋊D3, C2×D3×#6258, … |
| maximal quotient of | C22×C3⋊#6258 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3007);