CoCalc -- 1117.sagews

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t := (1,2,3,4,5,6,7,8,9,10,11,12);

(1,2,3,4,5,6,7,8,9,10,11,12)

t*t;

(1,3,5,7,9,11)(2,4,6,8,10,12)

t^2;

(1,3,5,7,9,11)(2,4,6,8,10,12)

t^-1;

(1,12,11,10,9,8,7,6,5,4,3,2)

(t^2)^-1;

(1,11,9,7,5,3)(2,12,10,8,6,4)

(t^2)^-1 = t^10;

true

T := Group(t);

Group([ (1,2,3,4,5,6,7,8,9,10,11,12) ])

IsomorphismGroups(T,CyclicGroup(12));

[ (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) ] -> 
[ f3, f1*f2 ]

Size(T);

12

List(T);

[ (), (1,9,5)(2,10,6)(3,11,7)(4,12,8), (1,5,9)(2,6,10)(3,7,11)(4,8,12), 
  (1,11,9,7,5,3)(2,12,10,8,6,4), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), 
  (1,3,5,7,9,11)(2,4,6,8,10,12), (1,12,11,10,9,8,7,6,5,4,3,2), 
  (1,8,3,10,5,12,7,2,9,4,11,6), (1,4,7,10)(2,5,8,11)(3,6,9,12), 
  (1,10,7,4)(2,11,8,5)(3,12,9,6), (1,6,11,4,9,2,7,12,5,10,3,8), 
  (1,2,3,4,5,6,7,8,9,10,11,12) ]

Orbit(T,1);

[ 1, 9, 5, 11, 7, 3, 12, 8, 4, 10, 6, 2 ]

IsTransitive(T);

true

Orbit(T,[4,7,12],OnSets);

[ [ 4, 7, 12 ], [ 4, 8, 11 ], [ 3, 8, 12 ], [ 2, 6, 9 ], [ 1, 6, 10 ], 
  [ 2, 5, 10 ], [ 1, 5, 8 ], [ 5, 9, 12 ], [ 1, 4, 9 ], [ 3, 7, 10 ], 
  [ 2, 7, 11 ], [ 3, 6, 11 ] ]

Orbit(T,[4,8,12],OnSets);

[ [ 4, 8, 12 ], [ 2, 6, 10 ], [ 1, 5, 9 ], [ 3, 7, 11 ] ]

i := (1,11)(2,10)(3,9)(4,8)(5,7);

(1,11)(2,10)(3,9)(4,8)(5,7)

TI := Group(t,i);

Group([ (1,2,3,4,5,6,7,8,9,10,11,12), (1,11)(2,10)(3,9)(4,8)(5,7) ])

IsomorphismGroups(TI,DihedralGroup(24));

[ (1,3)(4,12)(5,11)(6,10)(7,9), (1,2)(3,12)(4,11)(5,10)(6,9)(7,8) ] -> 
[ f1, f1*f2 ]

m := (1,5)(2,10)(4,8)(7,11);

(1,5)(2,10)(4,8)(7,11)

TTO := Group(t,i,m);

Group([ (1,2,3,4,5,6,7,8,9,10,11,12), (1,11)(2,10)(3,9)(4,8)(5,7), 
  (1,5)(2,10)(4,8)(7,11) ])

IsomorphismGroups(TTO,DirectProduct(DihedralGroup(6),DihedralGroup(8)));

[ (1,5)(2,4)(6,12)(7,11)(8,10), (1,4,7,10)(2,5,8,11)(3,6,9,12), 
  (2,6)(3,11)(5,9)(8,12) ] -> [ f1*f2^2*f3, f4, f1 ]

t^2; i*t^3; i*t^5;

(1,3,5,7,9,11)(2,4,6,8,10,12)
(1,2)(3,12)(4,11)(5,10)(6,9)(7,8)
(1,4)(2,3)(5,12)(6,11)(7,10)(8,9)

AutTI := AutomorphismGroup(TI);

<group of size 48 with 5 generators>

alpha := IsomorphismGroups(TTO,AutTI);

[ (1,4,7,10)(2,9,8,3)(5,12,11,6), (1,3)(4,12)(5,11)(6,10)(7,9), 
  (1,4,7,10)(2,5,8,11)(3,6,9,12) ] -> 
[ [ (1,9)(2,8)(3,7)(4,6)(10,12), (1,10,7,4)(2,11,8,5)(3,12,9,6), 
      (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,9,5)(2,10,6)(3,11,7)(4,12,8) ] -> 
    [ (1,6)(2,5)(3,4)(7,12)(8,11)(9,10), (1,10,7,4)(2,11,8,5)(3,12,9,6), 
      (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12) ], 
  [ (1,2,3,4,5,6,7,8,9,10,11,12), (1,11)(2,10)(3,9)(4,8)(5,7) ] -> 
    [ (1,12,11,10,9,8,7,6,5,4,3,2), (1,3)(4,12)(5,11)(6,10)(7,9) ], 
  [ (1,2,3,4,5,6,7,8,9,10,11,12), (1,11)(2,10)(3,9)(4,8)(5,7) ] -> 
    [ (1,2,3,4,5,6,7,8,9,10,11,12), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11) ] ]

HypTI := Group(t^alpha,i^alpha);

<group with 2 generators>

hyp_t := (t^7)^alpha;

[ (1,8)(2,7)(3,6)(4,5)(9,12)(10,11), (1,10,7,4)(2,11,8,5)(3,12,9,6), 
  (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,9,5)(2,10,6)(3,11,7)(4,12,8) ] -> 
[ (1,9)(2,8)(3,7)(4,6)(10,12), (1,10,7,4)(2,11,8,5)(3,12,9,6), 
  (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,9,5)(2,10,6)(3,11,7)(4,12,8) ]

hyp_i := i^alpha;

[ (2,12)(3,11)(4,10)(5,9)(6,8), (1,4,7,10)(2,5,8,11)(3,6,9,12), 
  (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12) ] -> 
[ (1,9)(2,8)(3,7)(4,6)(10,12), (1,10,7,4)(2,11,8,5)(3,12,9,6), 
  (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,9,5)(2,10,6)(3,11,7)(4,12,8) ]

(t^2)^(hyp_t^2); (i*t^3)^(hyp_t^2); (i*t^5)^(hyp_t^2);

(1,3,5,7,9,11)(2,4,6,8,10,12)
(1,4)(2,3)(5,12)(6,11)(7,10)(8,9)
(1,6)(2,5)(3,4)(7,12)(8,11)(9,10)

(t^2)^(hyp_i*hyp_t^3);

(1,11,9,7,5,3)(2,12,10,8,6,4)

(i*t^3)^(hyp_i*hyp_t^3);

(1,11)(2,10)(3,9)(4,8)(5,7)

(i*t^5)^(hyp_i*hyp_t^3);

(1,9)(2,8)(3,7)(4,6)(10,12)

(t^2)^(((hyp_t^2)^-1)*hyp_i*hyp_t^3);

(1,11,9,7,5,3)(2,12,10,8,6,4)

(i*t^5)^(((hyp_t^2)^-1)*hyp_i*hyp_t^3);

(1,11)(2,10)(3,9)(4,8)(5,7)

(i*t^7)^(((hyp_t^2)^-1)*hyp_i*hyp_t^3);

(1,9)(2,8)(3,7)(4,6)(10,12)

((hyp_t^2)^-1)*hyp_i*hyp_t^3 = hyp_i*hyp_t^5;

true

hyp_m := m^alpha;

[ (1,6,11,4,9,2,7,12,5,10,3,8), (1,11)(2,10)(3,9)(4,8)(5,7) ] -> 
[ (1,2,3,4,5,6,7,8,9,10,11,12), (1,11)(2,10)(3,9)(4,8)(5,7) ]

hyp_mi := hyp_m*hyp_i;

[ (1,6,11,4,9,2,7,12,5,10,3,8), (1,11)(2,10)(3,9)(4,8)(5,7) ] -> 
[ (1,12,11,10,9,8,7,6,5,4,3,2), (1,11)(2,10)(3,9)(4,8)(5,7) ]

InnTI := InnerAutomorphismsAutomorphismGroup(AutTI);

<group with 2 generators>

(t^2)^(t^5); (i*t^5)^(t^5); (i*t^7)^(t^5);

(1,3,5,7,9,11)(2,4,6,8,10,12)
(1,2)(3,12)(4,11)(5,10)(6,9)(7,8)
(1,4)(2,3)(5,12)(6,11)(7,10)(8,9)

beta := ActionHomomorphism(HypTI,InnTI);

<action homomorphism>

ker_beta := Kernel(beta);

<group with 1 generators>

Size(ker_beta);

2

List(ker_beta);

[ IdentityMapping( Group([ (1,2,3,4,5,6,7,8,9,10,11,12), 
      (1,11)(2,10)(3,9)(4,8)(5,7) ]) ), 
  [ (1,9)(2,8)(3,7)(4,6)(10,12), (1,10,7,4)(2,11,8,5)(3,12,9,6), 
      (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,9,5)(2,10,6)(3,11,7)(4,12,8) ] -> 
    [ (1,3)(4,12)(5,11)(6,10)(7,9), (1,10,7,4)(2,11,8,5)(3,12,9,6), 
      (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,9,5)(2,10,6)(3,11,7)(4,12,8) ] ]

ker_beta = Center(HypTI);

true

t^(t^0) = t^(t^6); i^(t^0) = i^(t^6);

true
true

t^(t^1) = t^(t^7); i^(t^1) = i^(t^7);

true
true

t^(i*t^4) = t^(i*t^10); i^(i*t^4) = i^(i*t^10);

true
true

CM := [4,7,12];

[ 4, 7, 12 ]

K := Orbit(TI,CM,OnSets);

[ [ 4, 7, 12 ], [ 4, 8, 11 ], [ 3, 8, 12 ], [ 2, 6, 9 ], [ 1, 6, 10 ], 
  [ 2, 5, 10 ], [ 1, 5, 8 ], [ 5, 9, 12 ], [ 1, 4, 9 ], [ 3, 7, 10 ], 
  [ 2, 7, 11 ], [ 3, 6, 11 ], [ 2, 7, 10 ], [ 3, 6, 10 ], [ 2, 6, 11 ], 
  [ 5, 8, 12 ], [ 1, 4, 8 ], [ 4, 9, 12 ], [ 1, 6, 9 ], [ 2, 5, 9 ], 
  [ 1, 5, 10 ], [ 4, 7, 11 ], [ 3, 7, 12 ], [ 3, 8, 11 ] ]

TI_K := Action(TI,K,OnSets);

Group([ (1,7,4,10,2,8,5,11,3,9,6,12)(13,24,18,21,15,23,17,20,14,22,16,19), 
  (1,16)(2,17)(3,18)(4,14)(5,15)(6,13)(7,22)(8,23)(9,24)(10,20)(11,21)(12,19) 
 ])

gamma := IsomorphismGroups(TI,TI_K);

[ (2,12)(3,11)(4,10)(5,9)(6,8), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11) ] -> 
[ (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24),
  (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,16)(8,17)(9,18)(10,14)(11,15)(12,13) 
 ]

t_K := Image(gamma,t);

(1,8,6,10,3,7,5,12,2,9,4,11)(13,23,16,21,14,24,17,19,15,22,18,20)

i_K := Image(gamma,i);

(1,18)(2,16)(3,17)(4,13)(5,14)(6,15)(7,24)(8,22)(9,23)(10,19)(11,20)(12,21)

S_24 := SymmetricGroup(24);

Sym( [ 1 .. 24 ] )

IsSubgroup(S_24,TI_K);

true

SW := Centralizer(S_24,TI_K);

<permutation group with 4 generators>

IsRegular(TI_K);

true

delta := IsomorphismGroups(TI_K,SW);

[ (1,22)(2,23)(3,24)(4,20)(5,21)(6,19)(7,14)(8,15)(9,13)(10,17)(11,18)(12,16),
  (1,7,4,10,2,8,5,11,3,9,6,12)(13,24,18,21,15,23,17,20,14,22,16,19) ] -> 
[ (1,13)(2,15)(3,14)(4,18)(5,17)(6,16)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19),
  (1,7,4,10,2,8,5,11,3,9,6,12)(13,19,16,22,14,20,17,23,15,21,18,24) ]

s := Image(delta,t_K);

(1,12,6,9,3,11,5,8,2,10,4,7)(13,24,18,21,15,23,17,20,14,22,16,19)

w := Image(delta,i_K);

(1,13)(2,15)(3,14)(4,18)(5,17)(6,16)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)

r := w;

(1,13)(2,15)(3,14)(4,18)(5,17)(6,16)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)

p := w*s;

(1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)

l := w*s^11;

(1,19)(2,21)(3,20)(4,24)(5,23)(6,22)(7,13)(8,15)(9,14)(10,18)(11,17)(12,16)

hp := w*s^5;

(1,23)(2,22)(3,24)(4,20)(5,19)(6,21)(7,17)(8,16)(9,18)(10,14)(11,13)(12,15)

T_K := Group(t_K);

Group([ (1,8,6,10,3,7,5,12,2,9,4,11)(13,23,16,21,14,24,17,19,15,22,18,20) ])

CinS_24ofT_K := Centralizer(S_24,T_K);

<permutation group with 5 generators>

Size(CinS_24ofT_K);

288

IsSubgroup(CinS_24ofT_K,SW);

true

minus := (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24);

(1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)

plus := minus^2;

()

t^plus;

(1,2,3,4,5,6,7,8,9,10,11,12)

t^minus;

(13,14,15,16,17,18,19,20,21,22,23,24)

d := (t^plus)^5*(t^minus)^5*plus;

(1,6,11,4,9,2,7,12,5,10,3,8)(13,18,23,16,21,14,19,24,17,22,15,20)

nhp := (t^plus)^3*(t^minus)^9*minus;

(1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)

QTT := WreathProduct(TI,SymmetricGroup(2));

<permutation group of size 1152 with 5 generators>

schritt := t^plus*(t^minus)^-1*plus;

(1,2,3,4,5,6,7,8,9,10,11,12)(13,24,23,22,21,20,19,18,17,16,15,14)

inversion := i^plus*i^minus*minus;

(1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(12,24)

SI := Group(schritt,inversion);

Group([ (1,2,3,4,5,6,7,8,9,10,11,12)(13,24,23,22,21,20,19,18,17,16,15,14), 
  (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(12,24) 
 ])

IsAbelian(SI);

true

i215 := m^plus*m^minus*minus;

(1,17)(2,22)(3,15)(4,20)(5,13)(6,18)(7,23)(8,16)(9,21)(10,14)(11,19)(12,24)

i216 := (m*i)^plus*(m*i)^minus*minus;

(1,19)(2,14)(3,21)(4,16)(5,23)(6,18)(7,13)(8,20)(9,15)(10,22)(11,17)(12,24)

i217 := i^plus*i^minus*(t^minus)^6*minus;

(1,23,7,17)(2,22,8,16)(3,21,9,15)(4,20,10,14)(5,19,11,13)(6,18,12,24)

i218 := (m*i)^plus*(m*i)^minus*(t^minus)^6*minus;

(1,19,7,13)(2,14,8,20)(3,21,9,15)(4,16,10,22)(5,23,11,17)(6,18,12,24)

i219 := m^plus*m^minus*(t^minus)^3*minus;

(1,17,4,20,7,23,10,14)(2,22,5,13,8,16,11,19)(3,15,6,18,9,21,12,24)

TI215 := Group(t^plus*t^minus,i215);

Group([ (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), 
  (1,17)(2,22)(3,15)(4,20)(5,13)(6,18)(7,23)(8,16)(9,21)(10,14)(11,19)(12,24) 
 ])

TI216 := Group(t^plus*t^minus,i216);

Group([ (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), 
  (1,19)(2,14)(3,21)(4,16)(5,23)(6,18)(7,13)(8,20)(9,15)(10,22)(11,17)(12,24) 
 ])

TI217 := Group(t^plus*t^minus,i217);

Group([ (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), 
  (1,23,7,17)(2,22,8,16)(3,21,9,15)(4,20,10,14)(5,19,11,13)(6,18,12,24) ])

TI218 := Group(t^plus*t^minus,i218);

Group([ (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), 
  (1,19,7,13)(2,14,8,20)(3,21,9,15)(4,16,10,22)(5,23,11,17)(6,18,12,24) ])

TI219 := Group(t^plus*t^minus,i219);

Group([ (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), 
  (1,17,4,20,7,23,10,14)(2,22,5,13,8,16,11,19)(3,15,6,18,9,21,12,24) ])

M := WreathProduct(TTO,SymmetricGroup(2));

<permutation group of size 4608 with 7 generators>