Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of (#4174⋊C2)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (#4174⋊C2)⋊C3 |
| unambiguous semidirect product | #92281⋊C2 |
| unambiguous semidirect product | (#4175⋊C3)⋊C2 |
| unambiguous semidirect product | (#4183⋊C3)⋊C2 |
| Fitting, Sylow | #6909 ⋊ C22 |
| derived | AΣL(1,𝔽27) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | AΣL(1,𝔽27) | C22×C3 |
| Frattini | C32 | C3×D32 |
| Fitting | #6909 | C22 |
| socle | C3 | C3×((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 | |
153 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 27 |
Polycyclic presentation:
2^2,3^5|e^a=e^2,f^a=f^2g,d^b=d^2,f^b=f^2,g^b=g^2,d^c=dg,e^d=ef,f^e=fg35 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 445);