Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of Dic36 |
| automorphism | Aut(G) = C22×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = D36 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C42×C9 ⋊ C2 |
| Sylow | C9 ⋊ #104 |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,5,7,9,8,3,6,4)(11,14,15,16);(2,4)(3,7)(5,6)(8,9)(10,11)(12,14)(13,15)(16,17)Polycyclic presentation:
2^5,3^2|b^2=c,d^2=e,f^3=g^2,b^a=bd,c^a=ce,d^a=de,f^a=f^2g,g^a=g^278 conjugacy classes — view the full character table.
| maximal subgroup of | #10588380, #10588397, D9×#104, … |
| maximal normal subgroup of | #10588380, #10588397, D9×#104, … |
| maximal quotient of | #10588128, #10588129, #10588133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 12);