Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| direct product | D3×Q8⋊H(3,𝔽3) |
| unambiguous semidirect product | C3⋊(C2×Q8⋊H(3,𝔽3)) |
| unambiguous semidirect product | (D3×Q8)⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×D3×SL(2,𝔽3))⋊C3 |
| Fitting | C33×Q8 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C3 | C3×D3×A4 |
| commutator, cosocle | C32×Q8 | C2×C32 |
| Fitting | C33×Q8 | C2×C3 |
| socle | C2×C32 | C2×C3×A4 |
| hypercenter | C2×C32 | D3×A4 |
| hypocenter | C3×Q8 | C2×H(3,𝔽3) |
| LCS | G ▷ C32×Q8 ▷ C3×Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32 | |
261 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
93 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2169);