Finite Group DatabasebetaFinite Group Databasebeta| order | 273 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = C2×C3×AGL(1,𝔽13) (order 936) · Inn(G) = C13⋊C3 · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C3×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×C13⋊C3 |
| unambiguous semidirect product | C13⋊(C3×C7) |
| Fitting, socle, Hall | C7×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7 | C13⋊C3 |
| commutator, cosocle, hypocenter | C13 | C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C7×C13 | C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C7 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 91 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4,5,6,7)(9,10,12)(11,14,17)(13,16,15)(18,20,19);(8,9,11,10,13,17,14,18,20,12,15,19,16)49 conjugacy classes — view the full character table.
| maximal subgroup of | C7×D13⋊C3, D7×C13⋊C3, (C13⋊C3)⋊D7, … |
| maximal normal subgroup of | C7×D13⋊C3, D7×C13⋊C3, (C13⋊C3)⋊D7, … |
| maximal quotient of | C2×C7×C13⋊C3, C7×C13⋊C9, C3×C7×C13⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(273, 2);