Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #3744 |
| automorphism | Aut(G) = AGL(1,ℤ/8)×AGL(2,𝔽3) (order 13824) · Inn(G) = C4⋊(C3⋊D3) · Out(G) = C22×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting | C32×C8 ⋊ C2 |
| Sylow | C32 ⋊ D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊(C3⋊D3) |
| commutator, cosocle | C32×C4 | C22 |
| Frattini | C4 | C2×C3⋊D3 |
| Fitting | C32×C8 | C2 |
| socle | C2×C32 | D4 |
| hypercenter | C8 | C3⋊D3 |
| hypocenter | C32 | D8 |
| LCS | G ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,6,8,4,7,3,5)(12,13,14);(2,5)(3,6)(7,8)(10,11)(13,14);(9,10,11)Polycyclic presentation:
2^4,3^2|b^2=cd,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 | #63546, #63549, #63626, … |
| maximal normal subgroup of | #63546, #63549, #63626, … |
| maximal quotient of | #63626, #63627, #63628, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 88);