Finite Group DatabasebetaFinite Group Databasebeta| order | 972 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C27 · Schur multiplier (H2) = C2×C33 |
| direct product | C3×A4×C27 |
| Fitting | C22×C3×C27 ⋊ C3 |
| derived, Sylow | C22 ⋊ C32×C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C27 | A4 |
| commutator, hypocenter | C22 | C32×C27 |
| Frattini | C9 | C32×A4 |
| Fitting | C22×C3×C27 | C3 |
| socle | C22×C32 | C3×C9 |
| prime radical | C32×A4 | C9 |
| cosocle | C22×C9 | C33 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C3×C27 | |
158 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
324 conjugacy classes. The character table is too large to display.
| derived subgroup of | C3⋊(A4⋊D27) |
| maximal subgroup of | C3×S4×C27, C27×C3⋊S4, C3×A4⋊D27, … |
| maximal normal subgroup of | C3×S4×C27, C27×C3⋊S4, C3×A4⋊D27, … |
| maximal quotient of | C3×SL(2,𝔽3)×C27, C2×C3×A4×C27 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(972, 518);