Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #157 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32×C4 · Schur multiplier (H2) = C23×C32 |
| direct product | C22×C32×#157 |
| Fitting, socle | C22×C34 ⋊ C4 |
| derived | C32 ⋊ C22×C32×C4 |
| Sylow | C34 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | #157 |
| commutator, hypocenter | C32 | C22×C32×C4 |
| Frattini | C1 | G |
| Fitting, socle | C22×C34 | C4 |
| prime radical | C22×C32×C3⋊D3 | C2 |
| cosocle | C3⋊D3 | C23×C32 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C22×C32 | |
736 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C32×PSU(3,𝔽2), C22×C3⋊D3×#157, C22×#10620209, … |
| maximal normal subgroup of | C22×C32×PSU(3,𝔽2), C22×C3⋊D3×#157, C22×#10620209, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3593);