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 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D9 |
| commutator, cosocle | C4×C9 | C23 |
| Frattini | C3×C4 | C22×D3 |
| Fitting | C9×SD16 | C2 |
| socle | C2×C3 | D3×D4 |
| hypercenter | SD16 | D9 |
| hypocenter | C9 | #135 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ SD16 | |
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:
(1,2,5,7,9,8,3,6,4)(10,12,14,20)(11,16,15,18)(13,22,21,19)(17,24,23,25);(11,15)(12,19)(13,21)(16,24)(18,25)(20,22);(2,4)(3,7)(5,6)(8,9)(10,11)(12,18)(13,17)(14,15)(16,20)(19,24)(21,23)(22,25)Polycyclic presentation:
2^5,3^2|c^2=e,d^2=e,f^3=g^2,b^a=be,c^a=ce,f^a=f^2g,g^a=g^2,c^b=cd,d^b=de,d^c=de42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63478, #10589864, D9×#135, … |
| maximal normal subgroup of | C2×#63478, #10589864, D9×#135, … |
| maximal quotient of | #10588436, #10588443, #10588448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 126);