Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #3714 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C32×#3714 |
| Sylow | C34 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | #626 |
| commutator, cosocle | C32×C4 | C22×C32 |
| Frattini | C4 | C32×D32 |
| Fitting | C34×C4 | C22 |
| socle | C2×C34 | D4 |
| hypercenter | C32×C4 | D32 |
| hypocenter | C32 | C32×D8 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2×C32 ◁ C32×C4 | |
588 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 588 conjugacy classes of subgroups — too many to draw. Only its 108 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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2715);