Spracherkennung für: .g vermutete Sprache: Unknown {[0] [0] [0]} [Methode: Schwerpunktbildung, einfache Gewichte, sechs Dimensionen]
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#V o8p2s3_o8p5s3_data
##
## In all representations (where applicable),
## $\langle a1, a2 \rangle \cong O^+_8(2)$,
## $\langle a1, a2, b \rangle \cong O^+_8(2).2$,
## $\langle a1, a2, t \rangle \cong O^+_8(2).3$,
## $\langle a1, a2, t, b \rangle \cong O^+_8(2).S_3$,
## $\langle a1, a2, c \rangle \cong O^+_8(5)$,
## $\langle a1, a2, c, b \rangle \cong O^+_8(5).2$,
## $\langle a1, a2, c, t \rangle \cong O^+_8(5).3$, and
## $\langle a1, a2, c, t, b \rangle \cong O^+_8(5).S_3$.
##
o8p2s3_o8p5s3_data:= rec(
# auxiliary data
o:= One( GF(5) ),
conj:= [[1,2,0,0,0,0,0,0],[4,2,0,0,0,0,0,0],
[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],
[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],
[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]*~.o,
conjinv:= Inverse( ~.conj ),
seed_orb120:= [1,1,0,0,0,0,0,0],
# matrix generators of $2.M$, $2.M.2$, in dimension $8$ over the Rationals
dim8Q:= rec(
a1:= [[-3,-1, 1, 1, 1,-1, 1, 1],[ 1,-1, 1, 1, 1,-1,-3, 1],
[-1, 1,-1,-1,-1, 1,-1, 3],[-1, 1,-1, 3,-1, 1,-1,-1],
[-1,-3,-1,-1,-1, 1,-1,-1],[-1, 1,-1,-1, 3, 1,-1,-1],
[ 1,-1,-3, 1, 1,-1, 1, 1],[ 1,-1, 1, 1, 1, 3, 1, 1]]/4,
a2:= [[ 0, 1, 0, 1, 0, 1, 0,-1],[-1, 0,-1, 0, 1, 0,-1, 0],
[-1, 0, 1, 0, 1, 0, 1, 0],[ 0,-1, 0,-1, 0, 1, 0,-1],
[ 0,-1, 0, 1, 0,-1, 0,-1],[ 1, 0,-1, 0, 1, 0, 1, 0],
[ 1, 0, 1, 0, 1, 0,-1, 0],[ 0, 1, 0,-1, 0,-1, 0,-1]]/2,
b:= ReflectionMat( ~.seed_orb120 ),
a1_t:= [[ 0, 0, 1, 1, 1, 1, 0, 0],[-1,-1, 0, 0, 0, 0,-1,-1],
[ 1,-1, 0, 0, 0, 0,-1, 1],[ 0, 0, 1,-1,-1, 1, 0, 0],
[ 0, 0, 1,-1, 1,-1, 0, 0],[ 1, 1, 0, 0, 0, 0,-1,-1],
[ 0, 0,-1,-1, 1, 1, 0, 0],[ 1,-1, 0, 0, 0, 0, 1,-1]]/2,
a2_t:= [[-1, 1,-1,-1, 1, 3, 1, 1],[ 1,-1,-3, 1,-1, 1,-1,-1],
[ 1, 3, 1, 1,-1, 1,-1,-1],[-1, 1,-1,-1, 1,-1,-3, 1],
[ 1,-1, 1, 1, 3, 1,-1,-1],[ 1,-1, 1,-3,-1, 1,-1,-1],
[ 1,-1, 1, 1,-1, 1,-1, 3],[-3,-1, 1, 1,-1, 1,-1,-1]]/4,
a1_tt:= [[-1,-1, 1, 1, 3,-1, 1, 1],[-1,-1,-3, 1,-1,-1, 1, 1],
[-1,-1, 1, 1,-1,-1,-3, 1],[-1,-1, 1, 1,-1,-1, 1,-3],
[ 1,-3,-1,-1, 1, 1,-1,-1],[ 3,-1, 1, 1,-1,-1, 1, 1],
[ 1, 1,-1, 3, 1, 1,-1,-1],[-1,-1, 1, 1,-1, 3, 1, 1]]/4,
a2_tt:= [[-1,-1, 3, 1,-1, 1,-1,-1],[ 1, 1, 1,-1, 1,-1, 1,-3],
[-3, 1, 1,-1, 1,-1, 1, 1],[-1,-1,-1,-3,-1, 1,-1,-1],
[ 1, 1, 1,-1, 1, 3, 1, 1],[-1,-1,-1, 1,-1, 1, 3,-1],
[ 1,-3, 1,-1, 1,-1, 1, 1],[-1,-1,-1, 1, 3, 1,-1,-1]]/4,
gamma:= [[ 0, 1, 0, 0, 0,-1,-1,-1],[-1, 0,-1, 1,-1, 0, 0, 0],
[ 1, 0,-1, 1, 1, 0, 0, 0],[ 0,-1, 0, 0, 0,-1, 1,-1],
[ 1, 0,-1,-1,-1, 0, 0, 0],[ 1, 0, 1, 1,-1, 0, 0, 0],
[ 0, 1, 0, 0, 0, 1, 1,-1],[ 0, 1, 0, 0, 0,-1, 1, 1]]/2,
gamma_t:= [[-3,-1,-1, 1, 1,-1,-1, 1],[-1,-3, 1,-1,-1, 1, 1,-1],
[ 1,-1,-1, 1,-3,-1,-1, 1],[ 1,-1, 3, 1, 1,-1,-1, 1],
[-1, 1, 1,-1,-1,-3, 1,-1],[-1, 1, 1,-1,-1, 1,-3,-1],
[ 1,-1,-1, 1, 1,-1,-1,-3],[-1, 1, 1, 3,-1, 1, 1,-1]]/4,
),
# permutation generators of $M$, $M.2$, of degree $120$
orb120:= SortedList( Orbit( Group( ~.dim8Q.a1, ~.dim8Q.a2 ),
~.seed_orb120, OnLines ) ),
deg120:= rec(
a1:= Permutation( ~.dim8Q.a1, ~.orb120, OnLines ),
a2:= Permutation( ~.dim8Q.a2, ~.orb120, OnLines ),
b:= Permutation( ~.dim8Q.b, ~.orb120, OnLines ),
a1_t:= Permutation( ~.dim8Q.a1_t, ~.orb120, OnLines ),
a2_t:= Permutation( ~.dim8Q.a2_t, ~.orb120, OnLines ),
a1_tt:= Permutation( ~.dim8Q.a1_tt, ~.orb120, OnLines ),
a2_tt:= Permutation( ~.dim8Q.a2_tt, ~.orb120, OnLines ),
gamma:= Permutation( ~.dim8Q.gamma, ~.orb120, OnLines ),
gamma_t:= Permutation( ~.dim8Q.gamma_t, ~.orb120, OnLines ),
),
# permutation generators of $M$, $M.2$, $M.3$, $M.S3$, of degree $360$
n:= 120,
t_3n:= PermList( Concatenation( [[1..~.n]+2*~.n,[1..~.n],[1..~.n]+~.n] ) ),
deg360:= rec(
a1:= ~.deg120.a1 * ( ~.deg120.a1_t^(~.t_3n) )
* ( ~.deg120.a1_tt^(~.t_3n^2) ),
a2:= ~.deg120.a2 * ( ~.deg120.a2_t^(~.t_3n) )
* ( ~.deg120.a2_tt^(~.t_3n^2) ),
b:= PermList( Concatenation( ListPerm( ~.deg120.b ),
ListPerm( ~.deg120.b * ~.deg120.gamma ) + 2*~.n,
ListPerm( ~.deg120.b * ~.deg120.gamma
* ~.deg120.gamma_t ) + ~.n ) ),
t:= ~.t_3n,
),
# matrix generators of $M$, $M.2$, $S$, $S.2$, in dimension $8$ over $\F_5$
dim8f5:= rec(
a1:= ~.conj * ~.dim8Q.a1 * ~.conjinv,
a2:= ~.conj * ~.dim8Q.a2 * ~.conjinv,
a1_t:= ~.conj * ~.dim8Q.a1_t * ~.conjinv,
a2_t:= ~.conj * ~.dim8Q.a2_t * ~.conjinv,
a1_tt:= ~.conj * ~.dim8Q.a1_tt * ~.conjinv,
a2_tt:= ~.conj * ~.dim8Q.a2_tt * ~.conjinv,
b:= ~.conj * ~.dim8Q.b * ~.conjinv,
c:= [[1,4,4,3,2,2,2,2],[4,1,0,3,1,4,3,1],
[0,3,0,4,2,1,1,3],[1,1,4,0,3,1,0,3],
[4,0,3,2,2,2,2,1],[4,3,1,1,3,0,3,2],
[0,0,4,0,1,3,2,1],[4,0,2,2,1,3,2,3]] * ~.o,
c_t:= [[3,3,3,1,0,4,1,3],[4,0,1,4,3,3,4,3],
[3,3,3,4,1,4,1,2],[3,2,4,0,0,0,3,0],
[3,1,2,0,0,2,0,0],[0,1,3,0,3,0,3,3],
[3,2,1,2,0,3,0,1],[4,1,0,0,1,2,4,1]] * ~.o,
c_tt:= [[4,4,2,4,2,4,0,3],[2,2,0,1,2,0,0,4],
[1,4,3,4,4,1,4,2],[1,3,0,3,1,4,4,4],
[1,1,4,1,0,0,2,3],[4,1,2,3,0,0,0,3],
[0,0,4,4,3,0,0,0],[2,4,1,3,3,2,0,0]] * ~.o,
gamma:= ~.conj * ~.dim8Q.gamma * ~.conjinv,
gamma_t:= ~.conj * ~.dim8Q.gamma_t * ~.conjinv,
),
# permutation generators of $M$, $M.2$, $S$, $S.2$, of degree $19\,656$
seed_orb19656:= [0,0,0,0,0,0,1,2] * ~.o,
orb19656:= SortedList( Orbit( Group( ~.dim8f5.a1, ~.dim8f5.a2, ~.dim8f5.c ),
~.seed_orb19656, OnLines ) ),
deg19656:= rec(
a1:= Permutation( ~.dim8f5.a1, ~.orb19656, OnLines ),
a2:= Permutation( ~.dim8f5.a2, ~.orb19656, OnLines ),
a1_t:= Permutation( ~.dim8f5.a1_t, ~.orb19656, OnLines ),
a2_t:= Permutation( ~.dim8f5.a2_t, ~.orb19656, OnLines ),
a1_tt:= Permutation( ~.dim8f5.a1_tt, ~.orb19656, OnLines ),
a2_tt:= Permutation( ~.dim8f5.a2_tt, ~.orb19656, OnLines ),
b:= Permutation( ~.dim8f5.b, ~.orb19656, OnLines ),
c:= Permutation( ~.dim8f5.c, ~.orb19656, OnLines ),
c_t:= Permutation( ~.dim8f5.c_t, ~.orb19656, OnLines ),
c_tt:= Permutation( ~.dim8f5.c_tt, ~.orb19656, OnLines ),
gamma:= Permutation( ~.dim8f5.gamma, ~.orb19656, OnLines ),
gamma_t:= Permutation( ~.dim8f5.gamma_t, ~.orb19656, OnLines ),
),
# permutation generators of $M$, $M.2$, $M.3$, $H$, $S$, $S.2$, $S.3$, $G$,
# of degree $58\,968$
N:= 19656,
t_3N:= PermList( Concatenation( [[1..~.N]+2*~.N,[1..~.N],[1..~.N]+~.N] ) ),
deg58968:= rec(
a1:= ~.deg19656.a1 * ( ~.deg19656.a1_t^~.t_3N )
* ( ~.deg19656.a1_tt^(~.t_3N^2) ),
a2:= ~.deg19656.a2 * ( ~.deg19656.a2_t^~.t_3N )
* ( ~.deg19656.a2_tt^(~.t_3N^2) ),
b:= PermList( Concatenation( ListPerm( ~.deg19656.b ),
ListPerm( ~.deg19656.b * ~.deg19656.gamma ) + 2*~.N,
ListPerm( ~.deg19656.b * ~.deg19656.gamma
* ~.deg19656.gamma_t ) + ~.N ) ),
t:= ~.t_3N,
c:= ~.deg19656.c * ( ~.deg19656.c_t^(~.t_3N) )
* ( ~.deg19656.c_tt^(~.t_3N^2) ),
),
# one block in the degree $58\,968$ permutation representation for $H$,
# which yields a degree $405$ representation of $H$
seed405:= [1,2,3,4,5,6,7,8,9,10,15,28,37,38,39,40,45,58,87,132,157,158,
159,160,165,178,207,252,397,662,807,4032,7257,7258,7259,7260,
7265,7278,7307,7352,7497,7762,8507,9732,13457,13458,13459,
13460,13461,13474,13487,13532,13577,13842,14107,15332],
# the permutation character of $G$ on the cosets of $H$, restricted to $H$,
# the ordering of classes corresponds to that in the GAP library table
# with identifier "O8+(2).3.2"
pi:= [51162109375,69375,1259375,69375,568750,1750,4000,375,135,975,135,625,
150,650,30,72,80,72,27,27,3,7,25,30,6,12,25,484375,1750,375,375,30,
40,15,15,15,6,6,3,3,3,157421875,121875,4875,475,75,3875,475,13000,
1750,300,400,30,60,15,15,15,125,10,30,4,8,6,9,7,5,6,5],
);;
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#E
