Polytope of Type {126}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {126}*252
Also Known As : 126-gon, {126}. if this polytope has another name.
Group : SmallGroup(252,14)
Rank : 2
Schlafli Type : {126}
Number of vertices, edges, etc : 126, 126
Order of s0s1 : 126
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{126,2} of size 504
{126,4} of size 1008
{126,4} of size 1008
{126,4} of size 1008
{126,6} of size 1512
{126,6} of size 1512
Vertex Figure Of :
{2,126} of size 504
{4,126} of size 1008
{4,126} of size 1008
{4,126} of size 1008
{6,126} of size 1512
{6,126} of size 1512
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {63}*126
3-fold quotients : {42}*84
6-fold quotients : {21}*42
7-fold quotients : {18}*36
9-fold quotients : {14}*28
14-fold quotients : {9}*18
18-fold quotients : {7}*14
21-fold quotients : {6}*12
42-fold quotients : {3}*6
63-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {252}*504
3-fold covers : {378}*756
4-fold covers : {504}*1008
5-fold covers : {630}*1260
6-fold covers : {756}*1512
7-fold covers : {882}*1764
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 19)( 5, 21)( 6, 20)( 7, 16)( 8, 18)( 9, 17)( 10, 13)
( 11, 15)( 12, 14)( 22, 44)( 23, 43)( 24, 45)( 25, 62)( 26, 61)( 27, 63)
( 28, 59)( 29, 58)( 30, 60)( 31, 56)( 32, 55)( 33, 57)( 34, 53)( 35, 52)
( 36, 54)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 65, 66)
( 67, 82)( 68, 84)( 69, 83)( 70, 79)( 71, 81)( 72, 80)( 73, 76)( 74, 78)
( 75, 77)( 85,107)( 86,106)( 87,108)( 88,125)( 89,124)( 90,126)( 91,122)
( 92,121)( 93,123)( 94,119)( 95,118)( 96,120)( 97,116)( 98,115)( 99,117)
(100,113)(101,112)(102,114)(103,110)(104,109)(105,111);;
s1 := ( 1, 88)( 2, 90)( 3, 89)( 4, 85)( 5, 87)( 6, 86)( 7,103)( 8,105)
( 9,104)( 10,100)( 11,102)( 12,101)( 13, 97)( 14, 99)( 15, 98)( 16, 94)
( 17, 96)( 18, 95)( 19, 91)( 20, 93)( 21, 92)( 22, 67)( 23, 69)( 24, 68)
( 25, 64)( 26, 66)( 27, 65)( 28, 82)( 29, 84)( 30, 83)( 31, 79)( 32, 81)
( 33, 80)( 34, 76)( 35, 78)( 36, 77)( 37, 73)( 38, 75)( 39, 74)( 40, 70)
( 41, 72)( 42, 71)( 43,110)( 44,109)( 45,111)( 46,107)( 47,106)( 48,108)
( 49,125)( 50,124)( 51,126)( 52,122)( 53,121)( 54,123)( 55,119)( 56,118)
( 57,120)( 58,116)( 59,115)( 60,117)( 61,113)( 62,112)( 63,114);;
poly := Group([s0,s1]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;; s1 := F.2;;
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(126)!( 2, 3)( 4, 19)( 5, 21)( 6, 20)( 7, 16)( 8, 18)( 9, 17)
( 10, 13)( 11, 15)( 12, 14)( 22, 44)( 23, 43)( 24, 45)( 25, 62)( 26, 61)
( 27, 63)( 28, 59)( 29, 58)( 30, 60)( 31, 56)( 32, 55)( 33, 57)( 34, 53)
( 35, 52)( 36, 54)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)
( 65, 66)( 67, 82)( 68, 84)( 69, 83)( 70, 79)( 71, 81)( 72, 80)( 73, 76)
( 74, 78)( 75, 77)( 85,107)( 86,106)( 87,108)( 88,125)( 89,124)( 90,126)
( 91,122)( 92,121)( 93,123)( 94,119)( 95,118)( 96,120)( 97,116)( 98,115)
( 99,117)(100,113)(101,112)(102,114)(103,110)(104,109)(105,111);
s1 := Sym(126)!( 1, 88)( 2, 90)( 3, 89)( 4, 85)( 5, 87)( 6, 86)( 7,103)
( 8,105)( 9,104)( 10,100)( 11,102)( 12,101)( 13, 97)( 14, 99)( 15, 98)
( 16, 94)( 17, 96)( 18, 95)( 19, 91)( 20, 93)( 21, 92)( 22, 67)( 23, 69)
( 24, 68)( 25, 64)( 26, 66)( 27, 65)( 28, 82)( 29, 84)( 30, 83)( 31, 79)
( 32, 81)( 33, 80)( 34, 76)( 35, 78)( 36, 77)( 37, 73)( 38, 75)( 39, 74)
( 40, 70)( 41, 72)( 42, 71)( 43,110)( 44,109)( 45,111)( 46,107)( 47,106)
( 48,108)( 49,125)( 50,124)( 51,126)( 52,122)( 53,121)( 54,123)( 55,119)
( 56,118)( 57,120)( 58,116)( 59,115)( 60,117)( 61,113)( 62,112)( 63,114);
poly := sub<Sym(126)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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