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) = C1 |
No aliases for this group.
| Sylow | C9 ⋊ #108 |
45 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,5,7,9,8,3,6,4)(10,12,13,21,15,22,14,23)(11,16,17,20,19,24,18,25);(2,4)(3,7)(5,6)(8,9)(10,11,14,18,15,19,13,17)(12,20,21,24,22,25,23,16)Polycyclic presentation:
2^5,3^2|a^2=d,b^2=c,c^2=e,d^2=e,f^3=g^2,b^a=bd,f^a=f^2g,g^a=g^2,d^b=de54 conjugacy classes — view the full character table.
| maximal subgroup of | #10588428, #10588429, #10588434, … |
| maximal normal subgroup of | #10588428, #10588429, #10588434, … |
| maximal quotient of | #10588156, #10588157, #10588231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 29);