Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of AΓL(1,𝔽9) |
| automorphism | Aut(G) = C22×AΓL(1,𝔽9) (order 576) · Inn(G) = AΓL(1,𝔽9) · Out(G) = C22 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | AGL(1,𝔽9)⋊C4 |
| Sylow | C32 ⋊ #106 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | AΓL(1,𝔽9) |
| commutator | #157 | C2×C4 |
| Fitting, socle | C2×C32 | SD16 |
| prime radical | C2×C3⋊D3 | D4 |
| cosocle | C2×#157 | C22 |
| hypocenter | C32 | #106 |
| LCS | G ▷ #157 ▷ C3⋊D3 ▷ C32 | |
| UCS | C1 ◁ C2 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,6,8)(2,5,7,4,9,3)(10,11,12,13);(2,3,4,8,5,7,9,6)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=d,c^2=e,b^a=bce,f^a=g,g^a=fg^2,c^b=ce,f^b=fg^2,g^b=g^2,f^c=fg^2,g^c=f^2g^2,f^e=f^2,g^e=g^218 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 843);