Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of (C3⋊#226)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| direct product | C3×(C3⋊GL(2,𝔽3))⋊C3 |
| unambiguous semidirect product | (C3⋊GL(2,𝔽3))⋊C32 |
| unambiguous semidirect product | (C3×C3⋊GL(2,𝔽3))⋊C3 |
| Fitting | C33×Q8 ⋊ D3 |
| derived | C3×SL(2,𝔽3) ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | (C3⋊S4)⋊C3 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C2×C32 |
| Frattini | C2×C3 | C32×S4 |
| Fitting | C33×Q8 | D3 |
| socle | C2×C32 | C3×S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
226 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2092);