Finite Group DatabasebetaFinite Group Databasebeta| order | 29760 |
| isoclinism | isoclinic class of Sp(2,𝔽31) |
| automorphism | Aut(G) = PGL(2,𝔽31) (order 29760) · Inn(G) = PSL(2,𝔽31) · Out(G) = C2 |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
| symplectic | Sp(2,𝔽31) |
| special linear | SL(2,𝔽31) |
| special unitary | SU(2,𝔽31) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C2 | PSL(2,𝔽31) |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
37 conjugacy classes of subgroups.
Permutation generators:
(1,2,4,3)(5,8,7,6)(9,21,15,22)(10,20,16,14)(11,19,17,13)(12,31,18,32)(23,43,41,26)(24,38,30,51)(25,53,42,54)(27,58,44,56)(28,34,45,40)(29,60,46,57)(33,37,47,50)(35,61,48,59)(36,62,49,55)(39,63,52,64);(2,5,9,23,55,57,46,17,44,16,43,53,38,40,42,22,31,48,30,28,59,50,19,49,20,52,63,58,33,12,6)(3,7,15,41,62,60,29,11,27,10,26,54,51,34,25,21,32,35,24,45,61,37,13,36,14,39,64,56,47,18,8);(1,4)(2,3)(5,10,28,32,59,39,44,26,33,62,57,25,9,24,17,7,16,45,31,61,52,27,43,47,55,60,42,15,30,11)(6,13,38,21,53,29,49,37,23,56,63,35,12,34,20,8,19,51,22,54,46,36,50,41,58,64,48,18,40,14)35 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,2,4,3)(5,8,7,6)(9,21,15,22)(10,20,16,14)(11,19,17,13)(12,31,18,32)(23,43,41,26)(24,38,30,51)(25,53,42,54)(27,58,44,56)(28,34,45,40)(29,60,46,57)(33,37,47,50)(35,61,48,59)(36,62,49,55)(39,63,52,64), (2,5,9,23,55,57,46,17,44,16,43,53,38,40,42,22,31,48,30,28,59,50,19,49,20,52,63,58,33,12,6)(3,7,15,41,62,60,29,11,27,10,26,54,51,34,25,21,32,35,24,45,61,37,13,36,14,39,64,56,47,18,8), (1,4)(2,3)(5,10,28,32,59,39,44,26,33,62,57,25,9,24,17,7,16,45,31,61,52,27,43,47,55,60,42,15,30,11)(6,13,38,21,53,29,49,37,23,56,63,35,12,34,20,8,19,51,22,54,46,36,50,41,58,64,48,18,40,14));