Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D3×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×C3×D3×#157 |
| Fitting, socle | C2×C34 ⋊ C2×C4 |
| derived | C33 ⋊ C22×C3×C4 |
| Sylow | C34 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D3×#157 |
| commutator, hypocenter | C33 | C22×C3×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C34 | C2×C4 |
| prime radical | C2×C3×D3×C3⋊D3 | C2 |
| cosocle | C3×C3⋊D3 | C23×C3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C2×C3 | |
478 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×D3×PSU(3,𝔽2), C2×D32×#157, C2×D3×C3⋊PSU(3,𝔽2), … |
| maximal normal subgroup of | C2×C3×D3×PSU(3,𝔽2), C2×D32×#157, C2×D3×C3⋊PSU(3,𝔽2), … |
| maximal quotient of | C3×#67×#157 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3564);