Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of #634⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C3 |
| direct product | C7×#634⋊C3 |
| unambiguous semidirect product | #634⋊(C3×C7) |
| Hall | C7×H(3,𝔽3) ⋊ Q8 |
| Sylow | H(3,𝔽3) ⋊ C7×Q8 |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 504 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2,3,4,5,6,7)(8,10,11,15)(9,12,13,14)(16,17,23)(18,20,22)(19,21,24);(8,9,11,13)(10,14,15,12)(17,19)(18,21)(20,22)(23,24)217 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 360);