Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of SU(3,𝔽2) |
| automorphism | Aut(G) = AGL(2,𝔽3) (order 432) · Inn(G) = PSU(3,𝔽2) · Out(G) = D3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| special unitary | SU(3,𝔽2) |
| Fitting, Sylow | H(3,𝔽3) ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C3 | PSU(3,𝔽2) |
| commutator, prime radical, cosocle | #272 | C22 |
| Fitting, hypocenter | H(3,𝔽3) | Q8 |
| LCS | G ▷ #272 ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,17,7,8,15,5,19,24,4,26,6,12)(2,21,9)(3,23,10,16,25,11,27,20,14,13,22,18);(2,5,6,16)(3,10,11,8)(7,20,21,26)(9,17,19,18)(12,14,22,23)(13,24,25,27)Polycyclic presentation:
2^3,3^3|a^2=c,b^2=c,b^a=bc,d^a=de^2,e^a=d^2e^2f^2,d^b=e,e^b=d^2f,d^c=d^2f,e^c=e^2f,e^d=ef16 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 88);