Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C22×#272 (order 216) · Inn(G) = C32 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C1 |
| direct product | C8×3−1+2 |
| unambiguous semidirect product | (C8×C9)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C32 |
| commutator | C3 | C32×C8 |
| Frattini, cosocle | C3×C4 | C2×C32 |
| socle | C2×C3 | C32×C4 |
| prime radical | C2×C32 | C3×C4 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ G | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,4,3,6,8,7,5,9)(10,11,12,14,13,15,16,17);(2,5,6)(4,8,9)88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 20);