Finite Group DatabasebetaFinite Group Databasebeta| order | 506 |
| isoclinism | isoclinic class of AGL(1,𝔽23) |
| automorphism | Aut(G) = AGL(1,𝔽23) (order 506) · Inn(G) = G · Out(G) = C1 |
| homology | Abelianization (H1) = C2×C11 · Schur multiplier (H2) = C1 |
| affine general linear | AGL(1,𝔽23) |
| unambiguous semidirect product | D23⋊C11 |
| unambiguous semidirect product | (C23⋊C11)⋊C2 |
| Fitting, socle, derived, Sylow | C23 ⋊ C2×C11 |
| Hall | C23⋊C11 ⋊ C2 |
| Hall | D23 ⋊ C11 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C23 | C2×C11 |
| LCS | G ▷ C23 | |
| UCS | C1 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 23 |
A minimal faithful permutation representation — 2 generators on 23 points:
(2,5,6,13,14,21,16,17,7,18,19,3,4,8,9,20,15,10,22,11,12,23);(1,2,4,7,6,10,12,20,9,15,17,23,19,22,21,13,14,18,16,8,11,5,3)Polycyclic presentation:
2,11,23|c^a=c^22,c^b=c^223 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(506, 1);