Finite Group DatabasebetaFinite Group Databasebeta| order | 12642 |
| isoclinism | isoclinic class of D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7×C43 · Schur multiplier (H2) = C1 |
| direct product | C7×D21×C43 |
| unambiguous semidirect product | C3⋊(C7×D7×C43) |
| unambiguous semidirect product | C7⋊(D3×C7×C43) |
| Fitting, socle, Hall | C3×C72×C43 ⋊ C2 |
| derived | C3×C7 ⋊ C2×C7×C43 |
| Hall | C72×C43 ⋊ D3 |
| Hall | C3×C72 ⋊ C2×C43 |
| Hall | C3×C43 ⋊ C7×D7 |
| Sylow | C72 ⋊ D3×C43 |
| Sylow | C3 ⋊ C7×D7×C43 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C7×C43 | D21 |
| commutator, cosocle, hypocenter | C3×C7 | C2×C7×C43 |
| Frattini | C1 | G |
| Fitting, socle | C3×C72×C43 | C2 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C7×C43 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 60 |
| transitive degree | μt(G) = 1806 |
A minimal faithful permutation representation — 2 generators on 60 points:
(1,2,3,4,5,6,7)(9,10)(11,12)(13,14)(16,17)(18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60);(8,9,11,13,14,12,10)(15,16,17)3612 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C7×D21×D43, D3×C7×D7×C43, C7×D21⋊D43, … |
| maximal normal subgroup of | C7×D21×D43, D3×C7×D7×C43, C7×D21⋊D43, … |
| maximal quotient of | C7×C43×Dic21, C2×C7×D21×C43, C7×C43×D63, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12642, 123);