Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = D4×D9 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C9×SD16 ⋊ C2 |
| Sylow | C9 ⋊ #137 |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,5,7,9,8,3,6,4)(11,16)(12,18)(13,21)(15,22)(19,20)(23,25);(10,12,14,20)(11,15,16,23)(13,18,21,19)(17,25,24,22);(2,4)(3,7)(5,6)(8,9)(10,11)(12,15)(13,17)(14,16)(18,25)(19,22)(20,23)(21,24)Polycyclic presentation:
2^5,3^2|b^2=e,d^2=e,f^3=g^2,c^a=ce,f^a=f^2g,g^a=g^2,c^b=cd,d^b=de,d^c=de39 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63477, #10589864, #10589870, … |
| maximal normal subgroup of | C2×#63477, #10589864, #10589870, … |
| maximal quotient of | #10588437, #10588439, #10588445, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 125);