Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of #625 |
| automorphism | Aut(G) = C23×D32 (order 288) · Inn(G) = D32 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D32 |
| commutator, socle | C2×C32 | C2×C4 |
| Fitting, cosocle | C32×C4 | C22 |
| prime radical | C2×C3×D3 | C4 |
| hypocenter | C32 | M16 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C4 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2,3,6,4,5,7,8)(10,11)(12,13,14);(2,5)(6,8)(9,10,11)(13,14)Polycyclic presentation:
2^4,3^2|b^2=c,c^2=d,b^a=bd,f^a=f^2,e^b=e^230 conjugacy classes — view the full character table.
| maximal subgroup of | D3×ΓL(1,𝔽25), #63791, #63803, … |
| maximal normal subgroup of | D3×ΓL(1,𝔽25), #63791, #63803, … |
| maximal quotient of | #63554, (C2×D3)⋊(C3⋊C8), Dic3⋊(C3⋊C8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 54);