Finite Group DatabasebetaFinite Group Databasebeta| order | 72 |
| isoclinism | isoclinic class of D9 |
| automorphism | Aut(G) = D4×AGL(1,ℤ/9) (order 432) · Inn(G) = D9 · Out(G) = C3×D4 |
| holomorph | G ⋊ Aut(G) = #72220184 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×Dic9 |
| unambiguous semidirect product | (C2×C9)⋊C4 |
| derived, Sylow | C9 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D9 |
| commutator, hypocenter | C9 | C2×C4 |
| Frattini | C2×C3 | C2×D3 |
| Fitting | C22×C9 | C2 |
| socle, prime radical | C22×C3 | D3 |
| cosocle | C2×C9 | C22 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C22 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(3,4,5,6)(8,10)(9,13)(11,12)(14,15);(1,2)(7,15,10,13,12,11,9,8,14)24 conjugacy classes — view the full character table.
| maximal subgroup of | C4×Dic9, Dic9⋊C4, C4⋊Dic9, … |
| maximal normal subgroup of | C4×Dic9, Dic9⋊C4, C4⋊Dic9, … |
| maximal quotient of | C2×C9⋊C8, #3667, C4×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(72, 7);