Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×#10619685 |
| Fitting | C23×#733 ⋊ C2 |
| Sylow | #733 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×#733⋊C2 |
| commutator, cosocle | C2×#733 | C23 |
| Frattini | C2×C32 | C22×C3⋊D3 |
| Fitting | C23×#733 | C2 |
| socle | C22×C3 | C2×#272 |
| hypercenter | C23 | #733⋊C2 |
| hypocenter | #733 | C2×D4 |
| LCS | G ▷ C2×#733 ▷ #733 | |
| UCS | C1 ◁ C22 ◁ C23 | |
351 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 351 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1200);