Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of ((C3⋊D3)⋊C3)⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | ((C3⋊D3)⋊C3)⋊S4 |
| Fitting | C22×H(3,𝔽3) ⋊ C2×D3 |
| derived | A4×H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | A4×H(3,𝔽3) | C22 |
| Frattini | C3 | D3×C3⋊S4 |
| Fitting | C22×H(3,𝔽3) | C2×D3 |
| socle | C22×C3 | D3×C3⋊D3 |
| LCS | G ▷ A4×H(3,𝔽3) | |
| UCS | C1 | |
363 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3^4,2^2|c^a=c^2,d^a=d^2,f^a=f^2,g^a=gh,e^b=e^2,f^b=f^2,g^c=h,h^c=gh,e^d=ef32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2931);