Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #1088⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#1088⋊C2 |
| unambiguous semidirect product | #1088⋊(C22×C3) |
| unambiguous semidirect product | (((C3⋊D3)⋊C3)⋊C2)⋊(C22×C3) |
| Fitting | C2×C3×H(3,𝔽3) ⋊ D4 |
| Sylow | C3×H(3,𝔽3) ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | #1088⋊C2 |
| commutator, cosocle | #272 | C23×C3 |
| Frattini | C3 | C2×C3×GO+(4,𝔽2) |
| Fitting | C2×C3×H(3,𝔽3) | D4 |
| socle | C2×C32 | GO+(4,𝔽2) |
| hypocenter | H(3,𝔽3) | C2×C3×D4 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
338 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 338 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2973);