Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #157 |
| automorphism | Aut(G) = C4×AΓL(1,𝔽9) (order 576) · Inn(G) = #157 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived, Sylow | C32 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | #157 |
| commutator, hypocenter | C32 | C16 |
| Fitting | C32×C4 | C4 |
| socle, prime radical | C2×C32 | C8 |
| cosocle | C3⋊(C3⋊C8) | C2 |
| LCS | G ▷ C32 | |
| UCS | C1 ◁ C4 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(3,5,6,4)(7,8,9,12,10,13,15,18,11,14,16,19,17,20,21,22);(1,3,6)(2,5,4)Polycyclic presentation:
2^4,3^2|a^2=b,b^2=c,c^2=d,e^a=ef^2,f^a=e^2f^2,e^b=e^2,f^b=f^224 conjugacy classes — view the full character table.
| derived subgroup of | #11772922, #11772938, #11772955, … |
| maximal subgroup of | #63725, #63734, #63735, … |
| maximal normal subgroup of | #63725, #63734, #63735, … |
| maximal quotient of | #63540, C2×#3708, #88410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 51);