Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of ((C3⋊S4)⋊C3)⋊D3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | ((C3⋊S4)⋊C3)⋊D3 |
| Fitting | C22×C33 ⋊ C2×D3 |
| derived | C3×C22⋊H(3,𝔽3) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C22⋊H(3,𝔽3) | C22 |
| Frattini | C3 | C3⋊D3×S4 |
| Fitting | C22×C33 | C2×D3 |
| socle | C22×C32 | D32 |
| LCS | G ▷ C3×C22⋊H(3,𝔽3) | |
| UCS | C1 | |
454 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 33 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|d^a=d^2,e^a=e^2,f^a=f^2,c^b=c^2,f^b=f^2,g^b=gh,e^c=ef,g^c=h,h^c=gh39 conjugacy classes — view the full character table.
| maximal subgroup of | C2×((C3⋊S4)⋊C3)⋊D3, #72222924, #72222956, … |
| maximal normal subgroup of | C2×((C3⋊S4)⋊C3)⋊D3, #72222924, D3×((C3⋊S4)⋊C3)⋊C2 |
| maximal quotient of | C2×((C3⋊S4)⋊C3)⋊D3, #72222951, #72222956 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2928);