Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of SL(2,𝔽3) |
| automorphism | Aut(G) = C22×C3×S4 (order 288) · Inn(G) = A4 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C4×C9 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #131⋊C9 |
| Sylow | #131 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | A4 |
| commutator, hypocenter | Q8 | C4×C9 |
| Frattini | C3×C4 | C2×A4 |
| Fitting | C3×#131 | C3 |
| socle | C2×C3 | C4×A4 |
| prime radical, cosocle | C3×#41 | C2×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C3×C8 | |
37 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,7,6,10,13,20)(4,9,25,18,17,23,26,32,14,8,12,29,15,24,27,30,5,21,11,19,16,22,28,31)(33,40,39,35,34,41,38,37,36);(1,14,6,16)(2,21,10,23)(3,25,13,27)(4,17,15,5)(7,29,20,31)(8,24,22,9)(11,28,26,12)(18,32,30,19)Polycyclic presentation:
2,3,2,3,2^3|a^2=c,b^3=d,c^2=g,e^2=g,f^2=g,e^b=f,f^b=ef,f^e=fg84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 77);