Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C32 |
| direct product | C4×C3⋊D32 |
| Fitting | C34×C4 ⋊ C22 |
| derived, Sylow | C34 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C3⋊D32 |
| commutator, hypocenter | C34 | C22×C4 |
| Frattini | C2 | C2×C3⋊D32 |
| Fitting | C34×C4 | C22 |
| socle, cosocle | C2×C34 | C23 |
| prime radical | C2×C3⋊D32 | C2 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C4 | |
1676 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 1676 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
144 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#10620215, C3⋊D32×AGL(1,𝔽5), C4×#10620198, … |
| maximal normal subgroup of | C4×#10620215, C4×#10620198, C4×#10620206 |
| maximal quotient of | C3⋊D32×AGL(1,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3410);