Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C3⋊((C3⋊D3)⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C32 |
| direct product | C23×C3⋊((C3⋊D3)⋊C3) |
| Fitting | C23×C3×H(3,𝔽3) ⋊ C2 |
| derived | C33 ⋊ C24×C3 |
| Sylow | C3×H(3,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | C3⋊((C3⋊D3)⋊C3) |
| commutator, cosocle, hypocenter | C33 | C24×C3 |
| Frattini | C3 | C23×C3×C3⋊D3 |
| Fitting | C23×C3×H(3,𝔽3) | C2 |
| socle | C23×C32 | C3×D3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C23 | |
1982 conjugacy classes of subgroups; only the 358 normal subgroups are shown.
This group has 1982 conjugacy classes of subgroups — too many to draw. Only its 358 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3070);