Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of C9⋊SL(2,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
| direct product | C2×C3×Q8⋊H(3,𝔽3) |
| unambiguous semidirect product | (C2×C3×Q8)⋊H(3,𝔽3) |
| Fitting | C2×C33×Q8 ⋊ C3 |
| Sylow | C2×Q8 ⋊ C3×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C3×A4 |
| commutator, cosocle | C3×Q8 | C2×C33 |
| Frattini | C2×C3 | C2×C32×A4 |
| Fitting | C2×C33×Q8 | C3 |
| hypercenter | C22×C33 | A4 |
| hypocenter | Q8 | C2×C3×H(3,𝔽3) |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C22×C32 ◁ C22×C33 | |
504 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 432 |
186 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2535);