Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C2×D4×GO+(4,𝔽2) (order 1152) · Inn(G) = C2×GO+(4,𝔽2) · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D32⋊Q8 |
| Sylow | C32 ⋊ C22⋊Q8 |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,3,4)(2,5,6)(7,9,10,14)(8,11,12,13);(3,4)(7,8,10,12)(9,13,14,11);(1,2)(3,6)(4,5)Polycyclic presentation:
2^5,3^2|b^2=e,c^2=e,b^a=bd,f^a=f^2,c^b=ce,f^b=g,g^b=f,f^d=f^2,g^d=g^227 conjugacy classes — view the full character table.
| maximal subgroup of | #10593284, #10593286, #10593288, … |
| maximal normal subgroup of | #10593284, #10593286, #10593288, … |
| maximal quotient of | #10593338, #10593340, #10593341, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 868);