Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3744 |
| automorphism | Aut(G) = C2×D4×AGL(2,𝔽3) (order 6912) · Inn(G) = C4⋊(C3⋊D3) · Out(G) = C2×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting | C32×C8 ⋊ C2 |
| Sylow | C32 ⋊ SD16 |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,3,5,4,6,7,8)(12,13,14);(2,5)(3,7)(6,8)(10,11)(13,14);(9,10,11)Polycyclic presentation:
2^4,3^2|b^2=c,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^239 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#204, #63795, #63801, … |
| maximal normal subgroup of | D3×#204, #63795, #63801, … |
| maximal quotient of | #63643, #63644, #63648, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 87);