Finite Group DatabasebetaFinite Group Databasebeta| order | 52 |
| isoclinism | isoclinic class of D13 |
| automorphism | Aut(G) = C2×AGL(1,𝔽13) (order 312) · Inn(G) = D13 · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = #72214210 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D26 |
| direct product | C2×D13 |
| unambiguous semidirect product | C13⋊C22 |
| unambiguous semidirect product | (C2×C13)⋊C2 |
| Fitting, socle | C2×C13 ⋊ C2 |
| derived, Sylow | C13 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D13 |
| commutator, cosocle, hypocenter | C13 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C13 | C2 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2 | |
10 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 26 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15);(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)16 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D13, D52, C22⋊D13, … |
| maximal normal subgroup of | C4×D13, D52, C22⋊D13, … |
| maximal quotient of | Dic26, C4×D13, D52, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(52, 4);