Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | Aut(G) = C22×AGL(2,𝔽3) (order 1728) · Inn(G) = C32 · Out(G) = C22×GL(2,𝔽3) |
| homology | Abelianization (H1) = C32×C8 · Schur multiplier (H2) = C32 |
| direct product | C8×H(3,𝔽3) |
| unambiguous semidirect product | (C32×C8)⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C8 | C32 |
| commutator | C3 | C32×C8 |
| Frattini, cosocle | C3×C4 | C2×C32 |
| socle | C2×C3 | C32×C4 |
| prime radical | C2×H(3,𝔽3) | C4 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C8 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,4)(3,6,8)(5,9,7)(10,11,12,14,13,15,16,17);(2,5,6)(4,8,9)88 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 19);