Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #64214 |
| automorphism | Aut(G) = D4×AΓL(1,𝔽9) (order 1152) · Inn(G) = C2×AGL(1,𝔽9) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊AGL(1,𝔽9) |
| Fitting | C32×C4 ⋊ C8 |
| Sylow | C32 ⋊ C4⋊C8 |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,4,9)(2,7,3)(5,8,6)(10,11,12,13);(2,3,4,7,5,8,9,6)(11,13)Polycyclic presentation:
2^5,3^2|a^2=c,b^2=d,c^2=e,b^a=bd,f^a=g,g^a=fg^2,f^c=fg^2,g^c=f^2g^2,f^e=f^2,g^e=g^224 conjugacy classes — view the full character table.
| maximal subgroup of | #10593155, #10593156, #10593157, … |
| maximal normal subgroup of | #10593155, #10593156, #10593157, … |
| maximal quotient of | #10593252, #10593253, #10593256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 864);