Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×D4×#272 (order 864) · Inn(G) = C1 · Out(G) = C2×D4×#272 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2×C3 |
| abelian | C2×C3×C4×C9 |
| multiplicative | (ℤ/247)ˣ, (ℤ/259)ˣ, (ℤ/333)ˣ, (ℤ/351)ˣ, (ℤ/494)ˣ, (ℤ/518)ˣ, (ℤ/666)ˣ, (ℤ/702)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C3 | C22×C32 |
| socle, prime radical | C22×C32 | C2×C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4)(7,8,9,10,11,12,13,14,15);(5,6)(16,17,18)216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 73);