Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #3672 |
| automorphism | Aut(G) = C22×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = #611 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C9×M16 ⋊ 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)(10,12,13,11,14,15,17,16);(2,4)(3,7)(5,6)(8,9)(10,11)(12,17)(13,15)(14,16)Polycyclic presentation:
2^5,3^2|b^2=c,c^2=e,d^2=e,f^3=g^2,b^a=bd,d^a=de,f^a=f^2g,g^a=g^2,d^b=de54 conjugacy classes — view the full character table.
| maximal subgroup of | #10588426, #10588427, #10588432, … |
| maximal normal subgroup of | #10588426, #10588427, #10588432, … |
| maximal quotient of | #10588141, #10588142, #10588144, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 32);