Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C4×D3 (order 96) · Inn(G) = D3 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C3×C16 · Schur multiplier (H2) = C1 |
| direct product | C3×C3⋊C16 |
| unambiguous semidirect product | C3⋊(C3×C16) |
| derived | C3 ⋊ C3×C16 |
| Sylow | C32 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C8 | D3 |
| commutator, hypocenter | C3 | C3×C16 |
| Frattini | C8 | C3×D3 |
| Fitting | C32×C8 | C2 |
| socle, prime radical | C2×C32 | C8 |
| cosocle | C3×C8 | C2×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C8 | |
24 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)(7,12,15,14,17,22,9,13,11,19,21,8,10,18,16,20);(4,5,6)72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×C3⋊C16, #63542, #63543, … |
| maximal normal subgroup of | D3×C3⋊C16, #63542, #63543, … |
| maximal quotient of | C3×C3⋊C32, C2×C3×C3⋊C16, C3×C9⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 28);