Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of AGL(1,ℤ/9) |
| automorphism | Aut(G) = D4×AGL(1,ℤ/9) (order 432) · Inn(G) = AGL(1,ℤ/9) · Out(G) = D4 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×3−1+2⋊C4 |
| unambiguous semidirect product | (C2×3−1+2)⋊C4 |
| unambiguous semidirect product | (C2×Dic9)⋊C3 |
| derived | C9 ⋊ C2×C3×C4 |
| Sylow | 3−1+2 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | AGL(1,ℤ/9) |
| commutator, hypocenter | C9 | C2×C3×C4 |
| Frattini | C2×C3 | C2×C3×D3 |
| Fitting | C22×3−1+2 | C2 |
| socle | C22×C3 | C3×D3 |
| prime radical | C22×C32 | D3 |
| cosocle | C2×C9 | C22×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C22 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 15 points:
(3,4,5,6)(8,12,15,10,11,14)(9,13);(1,2)(7,15,10,13,11,12,9,8,14)40 conjugacy classes — view the full character table.
| maximal subgroup of | C4×3−1+2⋊C4, (3−1+2⋊C4)⋊C4, C4⋊(3−1+2⋊C4), … |
| maximal normal subgroup of | C4×3−1+2⋊C4, (3−1+2⋊C4)⋊C4, C4⋊(3−1+2⋊C4), … |
| maximal quotient of | C2×3−1+2⋊C8, #3667⋊C3, C4×3−1+2⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 61);