Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| automorphism | Aut(G) = C2×D32×D4 (order 576) · Inn(G) = C2×D32 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | D3×D12 |
| Fitting | C32×C4 ⋊ C22 |
| Sylow | C32 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32 |
| commutator, socle, cosocle | C2×C32 | C23 |
| Fitting | C32×C4 | C22 |
| hypercenter | C4 | D32 |
| hypocenter | C32 | C2×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 | |
116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,2,3,4)(5,6,7)(8,9,10);(2,4)(9,10);(6,7)27 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#204, D3×D24, #63794, … |
| maximal normal subgroup of | D3×#204, D3×D24, #63794, … |
| maximal quotient of | D3×#204, D3×D24, #63794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 144);