Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = D4×#272 (order 432) · Inn(G) = C32 · Out(G) = D3×D4 |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C4×3−1+2 |
| unambiguous semidirect product | (C2×C4×C9)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C32 |
| commutator | C3 | C2×C32×C4 |
| Frattini, cosocle | C2×C3 | C22×C32 |
| socle | C22×C3 | C2×C32 |
| prime radical | C22×C32 | C2×C3 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(7,8,10,9,12,14,13,11,15);(3,4,5,6)(8,11,12)(10,14,15)88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 75);