Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | D4×#4169 |
| Fitting | D4×C32⋊C9 ⋊ C2 |
| Sylow | C32⋊C9 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×#4169 |
| commutator, cosocle | C2×C3×C9 | C23×C3 |
| Frattini, socle | C2×C32 | C22×C3×D3 |
| Fitting | D4×C32⋊C9 | C2 |
| hypercenter | D4 | #4169 |
| hypocenter | C3×C9 | C2×C3×D4 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ D4 | |
437 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#4169⋊C3, D4×((C3⋊D3)⋊C9)⋊C2, C2×D4×#4169, … |
| maximal normal subgroup of | D4×#4169⋊C3, D4×((C3⋊D3)⋊C9)⋊C2, C2×D4×#4169 |
| maximal quotient of | C2×D4×#4169 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1057);