Finite Group DatabasebetaFinite Group Databasebeta| order | 12144 |
| isoclinism | isoclinic class of Sp(2,𝔽23) |
| automorphism | Aut(G) = PGL(2,𝔽23) (order 12144) · Inn(G) = PSL(2,𝔽23) · Out(G) = C2 |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
| symplectic | Sp(2,𝔽23) |
| special linear | SL(2,𝔽23) |
| special unitary | SU(2,𝔽23) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C2 | PSL(2,𝔽23) |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
28 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)(12,30,18,31)(23,43,37,33)(24,46,38,45)(25,42,39,32)(26,44,27,36)(28,48,40,47)(29,35,41,34);(2,5,9,16,39,36,37,40,17,26,46,22,30,47,34,20,38,42,29,43,19,12,6)(3,7,15,10,25,44,23,28,11,27,45,21,31,48,35,14,24,32,41,33,13,18,8);(1,4)(2,3)(5,10,26,25,30,48,38,15,37,41,17,7,16,27,39,31,47,24,9,23,29,11)(6,13,34,33,22,45,40,18,42,44,20,8,19,35,43,21,46,28,12,32,36,14)27 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)(12,30,18,31)(23,43,37,33)(24,46,38,45)(25,42,39,32)(26,44,27,36)(28,48,40,47)(29,35,41,34), (2,5,9,16,39,36,37,40,17,26,46,22,30,47,34,20,38,42,29,43,19,12,6)(3,7,15,10,25,44,23,28,11,27,45,21,31,48,35,14,24,32,41,33,13,18,8), (1,4)(2,3)(5,10,26,25,30,48,38,15,37,41,17,7,16,27,39,31,47,24,9,23,29,11)(6,13,34,33,22,45,40,18,42,44,20,8,19,35,43,21,46,28,12,32,36,14));