Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| automorphism | Aut(G) = C22×D4×AGL(1,ℤ/9) (order 1728) · Inn(G) = C22×D9 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C9×M16 ⋊ C2 |
| Sylow | C9 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D9 |
| commutator | C2×C9 | C22×C4 |
| Frattini | C3×C4 | C22×D3 |
| Fitting | C9×M16 | C2 |
| socle | C2×C3 | C2×C4×D3 |
| prime radical | #3697 | C2 |
| cosocle | C4×C9 | C23 |
| hypercenter | M16 | D9 |
| hypocenter | C9 | #131 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C4 ◁ M16 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,6,8,7,2,4,3,9,5)(10,12,13,19,14,18,20,24)(11,16,17,23,15,21,22,25);(11,15)(12,18)(17,22)(19,24);(2,4)(3,7)(5,6)(8,9)(10,11)(12,16)(13,17)(14,15)(18,21)(19,23)(20,22)(24,25)Polycyclic presentation:
2^5,3^2|c^2=d,d^2=e,f^3=g^2,b^a=be,f^a=f^2g,g^a=g^2,c^b=ce60 conjugacy classes — view the full character table.
| maximal subgroup of | #10588428, #10588429, #10588434, … |
| maximal normal subgroup of | #10588428, #10588429, #10588434, … |
| maximal quotient of | #10588381, #10588387, #10588389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 117);