Finite Group DatabasebetaFinite Group Databasebeta| order | 44 |
| isoclinism | isoclinic class of D11 |
| automorphism | Aut(G) = C2×AGL(1,𝔽11) (order 220) · Inn(G) = D11 · Out(G) = C2×C5 |
| holomorph | G ⋊ Aut(G) = #72213693 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D22 |
| direct product | C2×D11 |
| unambiguous semidirect product | C11⋊C22 |
| unambiguous semidirect product | (C2×C11)⋊C2 |
| Fitting, socle | C2×C11 ⋊ C2 |
| derived, Sylow | C11 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D11 |
| commutator, cosocle, hypocenter | C11 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C11 | C2 |
| LCS | G ▷ C11 | |
| UCS | C1 ◁ C2 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 22 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2)(3,4,5,6,7,8,9,10,11,12,13);(4,13)(5,12)(6,11)(7,10)(8,9)14 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D11, D44, C22⋊D11, … |
| maximal normal subgroup of | C4×D11, D44, C22⋊D11, … |
| maximal quotient of | Dic22, C4×D11, D44, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(44, 3);