Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×D3×B(4,𝔽2) (order 768) · Inn(G) = C2×D3 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×D12 |
| Fitting | C2×C32×C4 ⋊ C2 |
| Sylow | C32 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3 | C2×D3 |
| commutator, cosocle | C2×C3 | C23×C3 |
| Frattini | C2 | C22×C3×D3 |
| Fitting | C2×C32×C4 | C2 |
| socle | C22×C32 | C22 |
| hypercenter | C2×C3×C4 | D3 |
| hypocenter | C3 | C2×C3×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4 | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 12 points:
(3,4,5)(6,7,8)(9,10,11,12);(1,2)(4,5)(10,12);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | #63558, #63562, #63563, … |
| maximal normal subgroup of | #63558, #63562, #63563, … |
| maximal quotient of | C3×#869, C3×C4×D12, C3×#874, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 160);