Finite Group DatabasebetaFinite Group Databasebeta| order | 444 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C4×D3×C9 (order 432) · Inn(G) = D3 · Out(G) = C2×C4×C9 |
| homology | Abelianization (H1) = C22×C37 · Schur multiplier (H2) = C2 |
| direct product | C2×D3×C37 |
| unambiguous semidirect product | C3⋊(C22×C37) |
| unambiguous semidirect product | (C2×C3)⋊(C2×C37) |
| Fitting, socle | C2×C3×C37 ⋊ C2 |
| derived, Sylow | C3 ⋊ C22×C37 |
| Hall | C3×C37 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C37 | D3 |
| commutator, cosocle, hypocenter | C3 | C22×C37 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C37 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C37 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 222 |
A minimal faithful permutation representation — 2 generators on 42 points:
(1,2)(4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42);(1,2)(3,4,5)222 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic37, #11725160, #11725162, … |
| maximal normal subgroup of | D3×Dic37, #11725160, #11725162, … |
| maximal quotient of | Dic6×C37, C4×D3×C37, D12×C37, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(444, 16);