Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #634 |
| automorphism | Aut(G) = C23×AGL(2,𝔽3) (order 3456) · Inn(G) = C2×C3⋊D3 · Out(G) = C2×GL(2,𝔽3) |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting | C32×C8 ⋊ C2 |
| Sylow | C32 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×C3⋊D3 |
| commutator, socle | C2×C32 | C2×C4 |
| Fitting | C32×C8 | C2 |
| prime radical | C2×C3⋊D3 | C4 |
| cosocle | C32×C4 | C22 |
| hypercenter | C8 | C3⋊D3 |
| hypocenter | C32 | M16 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 ◁ C8 | |
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:
(1,2,3,6,4,5,7,8)(12,13,14);(2,5)(6,8)(10,11)(13,14);(9,10,11)Polycyclic presentation:
2^4,3^2|b^2=c,c^2=d,b^a=bd,e^a=e^2,f^a=f^242 conjugacy classes — view the full character table.
| maximal subgroup of | D3×ΓL(1,𝔽25), #63798, #63804, … |
| maximal normal subgroup of | D3×ΓL(1,𝔽25), #63798, #63804, … |
| maximal quotient of | #63641, #63642, #63647, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 86);