Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 18144 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C23×C3×C9 |
| multiplicative | (ℤ/399)ˣ, (ℤ/532)ˣ, (ℤ/684)ˣ, (ℤ/756)ˣ, (ℤ/798)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C23×C32 |
| socle, prime radical | C23×C32 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(7,8,9,10,11,12,13,14,15);(3,4)(16,17,18);(5,6)216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 114);