Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×D3×GL(2,𝔽3) (order 1152) · Inn(G) = D3 · Out(G) = C22×GL(2,𝔽3) |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C3 |
| direct product | C32×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C32×C8) |
| derived | C3 ⋊ C32×C8 |
| Sylow | C33 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C4 | D3 |
| commutator, hypocenter | C3 | C32×C8 |
| Frattini | C4 | C32×D3 |
| Fitting | C33×C4 | C2 |
| socle, prime radical | C2×C33 | C4 |
| cosocle | C3×C4 | C2×C32 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C32×C4 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(4,5,6)(8,9)(10,14,16,11,13,17,12,15);(1,2,3)(7,8,9)108 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×C3⋊C8, C3×#3710, C3×#3711, … |
| maximal normal subgroup of | C3×D3×C3⋊C8, C3×#3710, C3×#3711, … |
| maximal quotient of | C32×C3⋊C16, C2×C32×C3⋊C8, C32×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 82);