Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C9×GL(3,𝔽2) (order 3024) · Inn(G) = C1 · Out(G) = C2×C9×GL(3,𝔽2) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C23×C27 |
| multiplicative | (ℤ/648)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C9 | C23×C3 |
| socle, prime radical | C23×C3 | C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33);(3,4);(5,6)216 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 24);