Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) order 72576 |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C32 |
| direct product | C23×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C32 |
| commutator, Frattini, cosocle | C3 | C23×C32 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2)(8,10,13)(11,15,14);(3,4)(7,8,11)(9,10,14)(12,13,15);(5,6)88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 115);