Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×D3×#120 (order 384) · Inn(G) = C2×D3 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#67 |
| Fitting | C23×C32 ⋊ 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 | C23×C32 | C2 |
| socle | C22×C32 | C22 |
| hypercenter | C23×C3 | D3 |
| hypocenter | C3 | C2×C3×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(3,4,5)(6,7,8)(10,12);(4,5)(9,10)(11,12);(1,2)54 conjugacy classes — view the full character table.
| maximal subgroup of | #63580, C3×#806, #63958, … |
| maximal normal subgroup of | #63580, C3×#806, #63958, … |
| maximal quotient of | C2×C3×Dic3⋊C4, C3×#924, C2×C3×(C2×D3)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 167);