Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9 · Schur multiplier (H2) = C2×C32 |
| direct product | A4×C27⋊C3 |
| Fitting | C22×C27⋊C3 ⋊ C3 |
| Sylow | C22 ⋊ C3×C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C9 | C32×A4 |
| commutator, socle | C22×C3 | C32×C9 |
| Fitting | C22×C27⋊C3 | C3 |
| prime radical | C32×A4 | C9 |
| cosocle | C22×C9 | C33 |
| hypercenter | C27⋊C3 | A4 |
| hypocenter | C22 | C3×C27⋊C3 |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C9 ◁ C27⋊C3 | |
132 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
132 conjugacy classes — view the full character table.
| maximal subgroup of | S4×C27⋊C3, A4⋊(D27⋊C3), A4×D27⋊C3, … |
| maximal normal subgroup of | S4×C27⋊C3, A4⋊(D27⋊C3), A4×D27⋊C3, … |
| maximal quotient of | SL(2,𝔽3)×C27⋊C3, C2×A4×C27⋊C3, #72229223 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 521);