Finite Group DatabasebetaFinite Group Databasebeta| order | 76 |
| isoclinism | isoclinic class of D19 |
| automorphism | Aut(G) = C2×AGL(1,𝔽19) (order 684) · Inn(G) = D19 · Out(G) = C2×C9 |
| holomorph | G ⋊ Aut(G) = #72227356 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D38 |
| direct product | C2×D19 |
| unambiguous semidirect product | C19⋊C22 |
| unambiguous semidirect product | (C2×C19)⋊C2 |
| Fitting, socle | C2×C19 ⋊ C2 |
| derived, Sylow | C19 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D19 |
| commutator, cosocle, hypocenter | C19 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C19 | C2 |
| LCS | G ▷ C19 | |
| UCS | C1 ◁ C2 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 38 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21);(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)22 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D19, D76, C22⋊D19, … |
| maximal normal subgroup of | C4×D19, D76, C22⋊D19, … |
| maximal quotient of | Dic38, C4×D19, D76, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(76, 3);