Finite Group DatabasebetaFinite Group Databasebeta| order | 546 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1 |
| direct product | C2×C7×C13⋊C3 |
| unambiguous semidirect product | (C2×C13)⋊(C3×C7) |
| Fitting, socle, Hall | C2×C7×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C2×C3×C7 |
| Hall | C7×C13 ⋊ C2×C3 |
| Hall | C2×C13 ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | C13⋊C3 |
| commutator, cosocle, hypocenter | C13 | C2×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C13 | C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2×C7 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 182 |
A minimal faithful permutation representation — 2 generators on 22 points:
(1,2)(3,4,5,6,7,8,9)(11,12,14)(13,16,19)(15,18,17)(20,22,21);(10,11,13,12,15,19,16,20,22,14,17,21,18)98 conjugacy classes — view the full character table.
| maximal subgroup of | C7×Dic13⋊C3, Dic7×C13⋊C3, #11763955, … |
| maximal normal subgroup of | C7×Dic13⋊C3, Dic7×C13⋊C3, #11763955, … |
| maximal quotient of | C4×C7×C13⋊C3, C22×C7×C13⋊C3, C2×C7×C13⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(546, 13);