Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = S4×AGL(1,ℤ/9) (order 1296) · Inn(G) = D9 · Out(G) = C3×S4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C22×D9 |
| unambiguous semidirect product | C9⋊C23 |
| unambiguous semidirect product | (C2×C9)⋊C22 |
| Fitting | C22×C9 ⋊ C2 |
| derived, Sylow | C9 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D9 |
| commutator, cosocle, hypocenter | C9 | C23 |
| Frattini | C3 | C22×D3 |
| Fitting | C22×C9 | C2 |
| socle | C22×C3 | D3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C22 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,2)(3,4)(6,8)(7,11)(9,12)(10,13);(3,4)(5,10,8,7,12,9,11,6,13);(1,2)24 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D9)⋊C4, C2×C4×D9, C2×D36, … |
| maximal normal subgroup of | (C2×D9)⋊C4, C2×C4×D9, C2×D36, … |
| maximal quotient of | C2×Dic18, C2×C4×D9, C2×D36, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 17);