Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| isoclinism | isoclinic class of A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C27 · Schur multiplier (H2) = C2 |
| direct product | C8×C22⋊C27 |
| unambiguous semidirect product | (C22×C8)⋊C27 |
| derived | C22 ⋊ C8×C27 |
| Sylow | C22×C8 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8×C9 | A4 |
| commutator, hypocenter | C22 | C8×C27 |
| Frattini | C4×C9 | C2×A4 |
| Fitting | C22×C8×C9 | C3 |
| socle, prime radical | C23×C3 | C4×C9 |
| cosocle | C22×C4×C9 | C2×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C8×C9 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 216 |
288 conjugacy classes — view the full character table.
| maximal subgroup of | (C22⋊C27)⋊C16, C16×C22⋊C27, #13136136, … |
| maximal normal subgroup of | (C22⋊C27)⋊C16, C16×C22⋊C27, #13136136, … |
| maximal quotient of | C8×Q8⋊C27, C16×C22⋊C27, #504⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 82);