Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of Dic18 |
| automorphism | Aut(G) = C25×AGL(1,ℤ/9) (order 1728) · Inn(G) = C2×D9 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×D9)⋊C8 |
| Fitting | C2×C8×C9 ⋊ C2 |
| Sylow | C9 ⋊ C22⋊C8 |
75 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,6,8,7,2,4,3,9,5)(11,13)(14,15,16,18,17,19,20,21);(2,4)(3,7)(5,6)(8,9)(10,11)(12,13)Polycyclic presentation:
2^5,3^2|b^2=c,c^2=e,f^3=g^2,b^a=bd,f^a=f^2g,g^a=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10588367, C8×D36, #10588370, … |
| maximal normal subgroup of | #10588367, C8×D36, #10588370, … |
| maximal quotient of | #10588133, #10588136, #10588145, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 27);