Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of #6229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×C4⋊#272 |
| unambiguous semidirect product | C4⋊(C7×#272) |
| Fitting | C4×C7×H(3,𝔽3) ⋊ C2 |
| Hall | C7×H(3,𝔽3) ⋊ D4 |
| Sylow | H(3,𝔽3) ⋊ C7×D4 |
176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 252 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4,5,6,7)(9,12,13)(11,15,16)(17,18,19,20);(9,11)(12,16)(13,15)(18,20);(8,9,11)(10,13,15)(12,16,14)217 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 399);