Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of C9⋊SL(2,𝔽3) |
| automorphism | Aut(G) = ((C3⋊D3)⋊C3)⋊S4 (order 1296) · Inn(G) = C3×A4 · Out(G) = D32 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | Q8⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×SL(2,𝔽3))⋊C3 |
| Fitting | C32×Q8 ⋊ C3 |
| Sylow | Q8 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2×C3 | C3×A4 |
| commutator, cosocle | C3×Q8 | C32 |
| Fitting | C32×Q8 | C3 |
| hypercenter | C2×C32 | A4 |
| hypocenter | Q8 | H(3,𝔽3) |
| LCS | G ▷ C3×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,7,4,5)(2,3,6,8)(10,13,14)(12,16,17);(2,3,5)(6,8,7)(9,10,12)(11,14,16)(13,17,15)Polycyclic presentation:
3^3,2^3|d^2=f,e^2=f,b^a=bc,d^a=e,e^a=de,e^d=ef31 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 42);