Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C23×SL(3,𝔽3) (order 44928) · Inn(G) = C1 · Out(G) = C23×SL(3,𝔽3) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C33 |
| abelian | C33×C8 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4 | C2×C33 |
| socle, prime radical | C2×C33 | C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3,4,5,6,7,8)(9,10,11);(12,13,14);(15,16,17)216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 85);