Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×#3811 |
| Fitting | C22×C34 ⋊ C22 |
| Sylow | C34 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C2×D32 |
| commutator, cosocle | C2×C32 | C23×C32 |
| Frattini | C2 | C2×C32×D32 |
| Fitting | C22×C34 | C22 |
| socle | C2×C34 | C23 |
| hypercenter | C22×C32 | D32 |
| hypocenter | C32 | C2×C32×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C32 ◁ C22×C32 | |
1140 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 204 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
216 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#3811, C2×C32×#3811, C3×#11724408, … |
| maximal normal subgroup of | C3⋊D3×#3811, C2×C32×#3811, C3×#11724408, … |
| maximal quotient of | C2×C32×#3811, 3−1+2×#3811, C3×#88668⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3339);