Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C2×#10596418 (order 1152) · Inn(G) = C2×PSU(3,𝔽2) · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | C22⋊PSU(3,𝔽2) |
| Fitting | C22×C32 ⋊ Q8 |
| Sylow | C32 ⋊ C22⋊Q8 |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 13 points:
(1,4,9)(2,7,3)(5,8,6)(11,13);(2,3,5,8)(4,6,9,7)(10,11)(12,13);(2,4,5,9)(3,7,8,6)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=d,b^a=bd,f^a=fg^2,g^a=f^2g^2,c^b=ce,f^b=g,g^b=f^2,f^d=f^2,g^d=g^218 conjugacy classes — view the full character table.
| maximal subgroup of | #10593165, #10596415, #10596416, … |
| maximal normal subgroup of | #10593165, #10596415, #10596416, … |
| maximal quotient of | #10593548, #10593550, #10593551, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 895);