Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C32×D4×D9 |
| Fitting | C32×D4×C9 ⋊ C2 |
| Sylow | C32×C9 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C22×D9 |
| commutator, cosocle | C2×C9 | C23×C32 |
| Frattini | C2×C3 | C22×C32×D3 |
| Fitting | C32×D4×C9 | C2 |
| socle | C2×C33 | C22×D3 |
| hypercenter | C32×D4 | D9 |
| hypocenter | C9 | C2×C32×D4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C32 ◁ C32×D4 | |
614 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
270 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D4×D9×C3⋊D3, C2×C32×D4×D9 |
| maximal normal subgroup of | D4×D9×C3⋊D3, C2×C32×D4×D9 |
| maximal quotient of | C2×C32×D4×D9 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2563);