Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of Dic18 |
| automorphism | Aut(G) = #120×AGL(1,ℤ/9) (order 1728) · Inn(G) = C2×D9 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C9 ⋊ C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D9 |
| commutator | C2×C9 | C2×C4 |
| Frattini, socle | C22×C3 | C2×D3 |
| Fitting | C23×C9 | C2 |
| prime radical | C23×C3 | D3 |
| cosocle | C22×C9 | C22 |
| hypercenter | C23 | D9 |
| hypocenter | C9 | C22⋊C4 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C23 | |
51 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,6,8,7,2,4,3,9,5)(15,17);(2,4)(3,7)(5,6)(8,9)(10,11,12,13)(14,15)(16,17)Polycyclic presentation:
2^4,3^2|a^2=c,e^3=f^2,b^a=bd,e^a=e^2f,f^a=f^242 conjugacy classes — view the full character table.
| maximal subgroup of | #63365, #63393, #63439, … |
| maximal normal subgroup of | #63365, #63393, #63439, … |
| maximal quotient of | #63389, C9⋊#95, #63391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 19);