Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #66419⋊C3 |
| Fitting, Sylow | 3−1+4 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C3 | C32×#157 |
| commutator, hypocenter | H(3,𝔽3) | C32×C4 |
| Fitting | 3−1+4 | C4 |
| prime radical | C3×#272 | C2×C3 |
| cosocle | #272 | C2×C32 |
| hypercenter | 3−1+2 | #157 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 ◁ 3−1+2 | |
120 conjugacy classes of subgroups.
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2,3^2,2,3^3|a^2=d,c^3=g^2,e^a=ef,f^a=ef^2g^2,c^b=cg^2,e^d=e^2,f^d=f^2g,f^e=fg62 conjugacy classes — view the full character table.
| maximal subgroup of | #10620020⋊C3, (#66419⋊C2)⋊C3, #14519150, … |
| maximal normal subgroup of | #10620020⋊C3, (#66419⋊C2)⋊C3, #14519150, … |
| maximal quotient of | #10619833⋊C3, C2×#66419⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 764);