Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×D3×#120 (order 384) · Inn(G) = D3 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
| direct product | C3×C4×Dic3 |
| unambiguous semidirect product | C3⋊(C3×C42) |
| derived | C3 ⋊ C3×C42 |
| Sylow | C32 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3×C4 | D3 |
| commutator, hypocenter | C3 | C3×C42 |
| Frattini | C22 | C2×C3×D3 |
| Fitting | C2×C32×C4 | C2 |
| socle, prime radical | C22×C32 | C22 |
| cosocle | C22×C3 | C22×C3 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C3×C4 | |
68 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)(5,6)(11,14,13,12);(4,5,6)(7,10,9,8)72 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×C3⋊C8, #63554, #63569, … |
| maximal normal subgroup of | Dic3×C3⋊C8, #63554, #63569, … |
| maximal quotient of | C3×C4×C3⋊C8, C3×#803, C3×C8×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 76);