Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of Dic18 |
| automorphism | Aut(G) = C22×C4×AGL(1,ℤ/9) (order 864) · Inn(G) = C2×D9 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C9×C16 ⋊ C2 |
| Sylow | C9 ⋊ M32 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D9 |
| commutator | C2×C9 | C2×C8 |
| Frattini | C3×C8 | C2×D3 |
| Fitting | C9×C16 | C2 |
| socle | C2×C3 | D3×C8 |
| prime radical | C2×D9 | C8 |
| cosocle | C8×C9 | C22 |
| hypercenter | C16 | D9 |
| hypocenter | C9 | M32 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C8 ◁ C16 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,6,8,7,2,4,3,9,5)(10,11,12,16,13,17,18,23,14,15,19,21,20,22,24,25);(2,4)(3,7)(5,6)(8,9)(11,15)(16,21)(17,22)(23,25)Polycyclic presentation:
2^5,3^2|b^2=c,c^2=d,d^2=e,f^3=g^2,b^a=be,f^a=f^2g,g^a=g^284 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#63357, #10588581, D9×M32, … |
| maximal normal subgroup of | C2×#63357, #10588581, D9×M32, … |
| maximal quotient of | Dic9⋊C16, #10588179, (C2×D9)⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 5);