Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C23×C3×D3 (order 144) · Inn(G) = D3 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C8×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C8×C9) |
| derived | C3 ⋊ C8×C9 |
| Sylow | C3×C9 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C9 | D3 |
| commutator, hypocenter | C3 | C8×C9 |
| Frattini | C3×C4 | C3×D3 |
| Fitting | C3×C4×C9 | C2 |
| socle, prime radical | C2×C32 | C3×C4 |
| cosocle | C32×C4 | C2×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C9 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3)(4,5,7,6,8,10,9,11,12)(13,14,15,17,16,18,19,20);(1,2,3)108 conjugacy classes — view the full character table.
| maximal subgroup of | D9×C3⋊C8, (C3⋊D9)⋊C8, #88413, … |
| maximal normal subgroup of | D9×C3⋊C8, (C3⋊D9)⋊C8, #88413, … |
| maximal quotient of | C9×C3⋊C16, C2×C9×C3⋊C8, C9×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 13);