Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3672 |
| automorphism | Aut(G) = C2×D4×AGL(1,ℤ/9) (order 864) · Inn(G) = #611 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | Q8×C9 ⋊ C2 |
| Sylow | C9 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #611 |
| commutator, cosocle | C4×C9 | C22 |
| Frattini | C3×C4 | C2×D3 |
| Fitting | Q8×C9 | C2 |
| socle | C2×C3 | #67 |
| prime radical | D36 | C2 |
| hypercenter | Q8 | D9 |
| hypocenter | C9 | SD16 |
| LCS | G ▷ C4×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,5,7,9,8,3,6,4)(10,11,13,15)(12,17,16,14);(2,4)(3,7)(5,6)(8,9)(11,14)(12,16)(15,17)Polycyclic presentation:
2^4,3^2|b^2=d,c^2=d,e^3=f^2,b^a=bc,c^a=cd,e^a=e^2f,f^a=f^2,c^b=cd27 conjugacy classes — view the full character table.
| maximal subgroup of | SD16×D9, #63476, #63480, … |
| maximal normal subgroup of | SD16×D9, #63476, #63480, … |
| maximal quotient of | #63367, #63369, #63395, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 18);