Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of #4174⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C22×#4174⋊C2 |
| unambiguous semidirect product | #4174⋊C23 |
| unambiguous semidirect product | #4175⋊C23 |
| unambiguous semidirect product | #4183⋊C23 |
| Fitting | C22×AΣL(1,𝔽27) ⋊ C22 |
| derived, Sylow | AΣL(1,𝔽27) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #4174⋊C2 |
| commutator, cosocle, hypocenter | AΣL(1,𝔽27) | C24 |
| Frattini | C32 | C22×D32 |
| Fitting | C22×AΣL(1,𝔽27) | C22 |
| socle | C22×C3 | ((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C22 | |
1104 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 1104 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) = 36 |
68 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#10620184⋊C2 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1860);