Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C22×C34 |
| direct product | A4×C22⋊H(3,𝔽3) |
| unambiguous semidirect product | (C22×A4)⋊H(3,𝔽3) |
| Fitting | C24×C32 ⋊ C32 |
| Sylow | C24 ⋊ C3×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×A42 |
| commutator, socle, cosocle | C24×C3 | C33 |
| Fitting | C24×C32 | C32 |
| hypercenter | C32 | A42 |
| hypocenter | C24 | C3×H(3,𝔽3) |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ C32 | |
354 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 354 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | A4×(C3⋊D3)⋊A4, #72220512, #72220518, … |
| maximal normal subgroup of | A4×(C3⋊D3)⋊A4, #72220512, #72220518, … |
| maximal quotient of | #72220499 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2963);