1. The generic table of cyclic groups.
For the cyclic group Cq = áx ñ of order q, there is one type of classes. The class parameters are integers k Î { 0, ¼, q-1 }, the class with parameter k consists of the group element xk. Group order and centralizer orders are the identity function q ® q, independent of the parameter k. The representative order function maps the parameter pair [q,k] to [(q)/(gcd(q,k))], which is the order of xk in Cq; the p-th power map is the function mapping the triple (q,k,p) to the parameter [1,(kp mod q)].
There is one type of characters with parameters l Î { 0, ¼, q-1 }; for eq a primitive complex q-th root of unity, the character values are cl(xk) = eqkl.
The library file contains the following generic table.
rec( identifier := "Cyclic", specializedname := ( q -> Concatenation( "C", String(q) ) ), size := ( n -> n ), text := "generic character table for cyclic groups", centralizers := [ function( n, k ) return n; end ], classparam := [ ( n -> [ 0 .. n-1 ] ) ], charparam := [ ( n -> [ 0 .. n-1 ] ) ], powermap := [ function( n, k, pow ) return [ 1, k*pow mod n ]; end ], orders := [ function( n, k ) return n / Gcd( n, k ); end ], irreducibles := [ [ function( n, k, l ) return E(n)^(k*l); end ] ], domain := IsPosInt, libinfo := rec( firstname:= "Cyclic", othernames:= [] ), isGenericTable := true )
2. The generic table of the general linear group GL(2,q).
We have four types t1, t2, t3, t4 of classes, according to the rational canonical form of the elements. t1 describes scalar matrices, t2 nonscalar diagonal matrices, t3 companion matrices of (X-r)2 for elements r Î \Fq*, and t4 companion matrices of irreducible polynomials of degree 2 over \Fq.
The sets of class parameters of the types are in bijection with \Fq* for t1 and t3, with the set {{r,t}; r, t Î \Fq*, r ¹ t} for t2, and with the set {{e,eq}; e Î \Fq2\\Fq} for t4.
The centralizer order functions are q ® (q2-1)(q2-q) for type t1, q ® (q-1)2 for type t2, q ® q(q-1) for type t3, and q ® q2-1 for type t4.
The representative order function of t1 maps (q,r) to the order of r in \Fq, that of t2 maps (q,{r,t}) to the least common multiple of the orders of r and t.
The file contains something similar to the following table.
rec(
identifier := "GL2",
specializedname := ( q -> Concatenation( "GL(2,", String(q), ")" ) ),
size := ( q -> (q^2-1)*(q^2-q) ),
text := "generic character table of GL(2,q), see Robert Steinberg: ...",
centralizers := [ function( q, k ) return (q^2-1) * (q^2-q); end,
..., ..., ... ],
classparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],
charparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],
powermap := [ function( q, k, pow ) return [ 1, (k*pow) mod (q-1) ]; end,
..., ..., ... ],
orders:= [ function( q, k ) return (q-1)/Gcd( q-1, k ); end,
..., ..., ... ],
irreducibles := [ [ function( q, k, l ) return E(q-1)^(2*k*l); end,
..., ..., ... ],
[ ..., ..., ..., ... ],
[ ..., ..., ..., ... ],
[ ..., ..., ..., ... ] ],
classtext := [ ..., ..., ..., ... ],
domain := ( q -> IsInt(q) and q > 1 and Length( Set( FactorsInt(q) ) ) = 1 ),
isGenericTable := true )
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GAP 4 manual