Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D32×#157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3 |
| direct product | D32×#157 |
| Fitting, socle, derived, Sylow | C34 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C34 | C22×C4 |
| prime radical | D32×C3⋊D3 | C2 |
| cosocle | C32×C3⋊D3 | C23 |
| LCS | G ▷ C34 | |
| UCS | C1 | |
478 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 478 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
54 conjugacy classes — view the full character table.
| maximal subgroup of | D32×PSU(3,𝔽2), D32×GO+(4,𝔽2), D32×AGL(1,𝔽9), … |
| maximal normal subgroup of | D32×PSU(3,𝔽2), D32×GO+(4,𝔽2), D32×AGL(1,𝔽9), … |
| maximal quotient of | #157×#624, D3×C3⋊D3×#157, C2×D32×#157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3524);