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×Q16 ⋊ C2 |
| Sylow | C9 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D9 |
| commutator, cosocle | C4×C9 | C23 |
| Frattini | C3×C4 | C22×D3 |
| Fitting | C9×Q16 | C2 |
| socle | C2×C3 | D3×D4 |
| prime radical | #3701 | C2 |
| hypercenter | Q16 | D9 |
| hypocenter | C9 | #137 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
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)(10,12,14,18)(11,15,17,24)(13,22,23,20)(16,25,21,19);(10,11,14,17)(12,19,18,25)(13,21,23,16)(15,22,24,20);(2,4)(3,7)(5,6)(8,9)(12,18)(15,24)(19,25)(20,22)Polycyclic presentation:
2^5,3^2|b^2=e,c^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×#63480, #10589868, #10589870, … |
| maximal normal subgroup of | C2×#63480, #10589868, #10589870, … |
| maximal quotient of | #10588469, #10588471, #10588474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 128);