Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of A4 |
| automorphism | Aut(G) = C3×D4×S4 (order 576) · Inn(G) = A4 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C2×C4×C9 · Schur multiplier (H2) = C22 |
| direct product | C2×C4×C22⋊C9 |
| unambiguous semidirect product | (C23×C4)⋊C9 |
| derived | C22 ⋊ C2×C4×C9 |
| Sylow | C23×C4 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | A4 |
| commutator, hypocenter | C22 | C2×C4×C9 |
| Frattini | C2×C3 | C22×A4 |
| Fitting | C23×C3×C4 | C3 |
| socle, prime radical | C24×C3 | C2×C3 |
| cosocle | C23×C3 | C22×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C3×C4 | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 19 points:
(4,5,6)(7,8,9,10)(11,18,17,13,12,19,16,15,14);(1,2)(3,6)(4,5)96 conjugacy classes — view the full character table.
| maximal subgroup of | C2×(C22⋊C9)⋊C8, #10589511, C4×(C22⋊C9)⋊C4, … |
| maximal normal subgroup of | C2×(C22⋊C9)⋊C8, #10589511, C4×(C22⋊C9)⋊C4, … |
| maximal quotient of | C42×C22⋊C9, C22⋊C4×C22⋊C9, C4⋊C4×C22⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 343);