Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C243 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C22⋊C243 |
| derived, Sylow | C22 ⋊ C243 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C81 | A4 |
| commutator, hypocenter | C22 | C243 |
| Fitting, cosocle | C22×C81 | C3 |
| socle, prime radical | C22×C3 | C81 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C81 | |
17 conjugacy classes of subgroups.
| transitive degree | μt(G) = 486 |
Polycyclic presentation:
3^5,2^2|a^3=b,b^3=c,c^3=d,d^3=e,f^a=fg,g^a=f324 conjugacy classes — view the full character table.
| derived subgroup of | (C22⋊C243)⋊C2 |
| maximal subgroup of | (C22⋊C243)⋊C2, C2×C22⋊C243 |
| maximal normal subgroup of | (C22⋊C243)⋊C2, C2×C22⋊C243 |
| maximal quotient of | Q8⋊C243, C2×C22⋊C243 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 3);