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×D8 ⋊ C2 |
| Sylow | C9 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D9 |
| commutator, cosocle | C4×C9 | C23 |
| Frattini | C3×C4 | C22×D3 |
| Fitting | C9×D8 | C2 |
| socle | C2×C3 | D3×D4 |
| hypercenter | D8 | D9 |
| hypocenter | C9 | AGL(1,ℤ/8) |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,5,7,9,8,3,6,4)(10,11)(12,17)(13,14)(15,16);(11,15)(12,16)(14,17);(2,4)(3,7)(5,6)(8,9)(11,14)(15,17)Polycyclic presentation:
2^5,3^2|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×#63473, #10589859, #10589870, … |
| maximal normal subgroup of | C2×#63473, #10589859, #10589870, … |
| maximal quotient of | #10588437, #10588441, #10588442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 121);