Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32 |
| direct product | D3×D4×H(3,𝔽3) |
| unambiguous semidirect product | (C32×D3×D4)⋊C3 |
| Fitting | C3×D4×H(3,𝔽3) ⋊ C2 |
| Sylow | C3×H(3,𝔽3) ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C3 | C22×C32×D3 |
| commutator, socle, cosocle | C2×C32 | C23×C32 |
| Fitting | C3×D4×H(3,𝔽3) | C2 |
| hypercenter | D4×H(3,𝔽3) | D3 |
| hypocenter | C3 | C2×D4×H(3,𝔽3) |
| LCS | G ▷ C2×C32 ▷ C3 | |
| UCS | C1 ◁ C2×C3 ◁ D4×H(3,𝔽3) | |
730 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 730 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
165 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D4×#272, D4×(C3⋊D3)⋊H(3,𝔽3) |
| maximal normal subgroup of | D3×D4×#272 |
| maximal quotient of | D4×(C3⋊D3)⋊H(3,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2569);