Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| automorphism | Aut(G) = C2×S4×AGL(1,ℤ/9) (order 2592) · Inn(G) = C22×D9 · Out(G) = C2×C3×D3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q8×C9 ⋊ C2 |
| Sylow | C9 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D9 |
| commutator, cosocle | C2×C9 | C23 |
| Frattini, socle | C2×C3 | C22×D3 |
| Fitting | Q8×C9 | C2 |
| hypercenter | Q8 | D9 |
| hypocenter | C9 | #41 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,6,8,7,2,4,3,9,5)(10,12,13,16)(11,14,15,17);(10,11,13,15)(12,17,16,14);(2,4)(3,7)(5,6)(8,9)(12,16)(14,17)Polycyclic presentation:
2^4,3^2|b^2=d,c^2=d,e^3=f^2,c^a=cd,e^a=e^2f,f^a=f^2,c^b=cd30 conjugacy classes — view the full character table.
| maximal subgroup of | #63476, #63478, #63480, … |
| maximal normal subgroup of | #63476, #63478, #63480, … |
| maximal quotient of | #63452, #63454, #63455, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 44);