Polytope of Type {2,36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,36,6}*864a
if this polytope has a name.
Group : SmallGroup(864,2437)
Rank : 4
Schlafli Type : {2,36,6}
Number of vertices, edges, etc : 2, 36, 108, 6
Order of s₀s₁s₂s₃ : 36
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,36,6,2} of size 1728
Vertex Figure Of :
   {2,2,36,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,18,6}*432a
   3-fold quotients : {2,36,2}*288, {2,12,6}*288a
   6-fold quotients : {2,18,2}*144, {2,6,6}*144a
   9-fold quotients : {2,12,2}*96, {2,4,6}*96a
   12-fold quotients : {2,9,2}*72
   18-fold quotients : {2,2,6}*48, {2,6,2}*48
   27-fold quotients : {2,4,2}*32
   36-fold quotients : {2,2,3}*24, {2,3,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,36,6}*1728a, {2,72,6}*1728a, {2,36,12}*1728a
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  7,  8)( 10, 11)( 12, 22)( 13, 21)( 14, 23)( 15, 25)( 16, 24)
( 17, 26)( 18, 28)( 19, 27)( 20, 29)( 31, 32)( 34, 35)( 37, 38)( 39, 49)
( 40, 48)( 41, 50)( 42, 52)( 43, 51)( 44, 53)( 45, 55)( 46, 54)( 47, 56)
( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)( 64, 92)
( 65, 91)( 66,103)( 67,102)( 68,104)( 69,106)( 70,105)( 71,107)( 72,109)
( 73,108)( 74,110)( 75, 94)( 76, 93)( 77, 95)( 78, 97)( 79, 96)( 80, 98)
( 81,100)( 82, 99)( 83,101);;
s2 := (  3, 66)(  4, 68)(  5, 67)(  6, 72)(  7, 74)(  8, 73)(  9, 69)( 10, 71)
( 11, 70)( 12, 57)( 13, 59)( 14, 58)( 15, 63)( 16, 65)( 17, 64)( 18, 60)
( 19, 62)( 20, 61)( 21, 76)( 22, 75)( 23, 77)( 24, 82)( 25, 81)( 26, 83)
( 27, 79)( 28, 78)( 29, 80)( 30, 93)( 31, 95)( 32, 94)( 33, 99)( 34,101)
( 35,100)( 36, 96)( 37, 98)( 38, 97)( 39, 84)( 40, 86)( 41, 85)( 42, 90)
( 43, 92)( 44, 91)( 45, 87)( 46, 89)( 47, 88)( 48,103)( 49,102)( 50,104)
( 51,109)( 52,108)( 53,110)( 54,106)( 55,105)( 56,107);;
s3 := (  3,  6)(  4,  7)(  5,  8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)( 22, 25)
( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)( 48, 51)
( 49, 52)( 50, 53)( 57, 60)( 58, 61)( 59, 62)( 66, 69)( 67, 70)( 68, 71)
( 75, 78)( 76, 79)( 77, 80)( 84, 87)( 85, 88)( 86, 89)( 93, 96)( 94, 97)
( 95, 98)(102,105)(103,106)(104,107);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(110)!(1,2);
s1 := Sym(110)!(  4,  5)(  7,  8)( 10, 11)( 12, 22)( 13, 21)( 14, 23)( 15, 25)
( 16, 24)( 17, 26)( 18, 28)( 19, 27)( 20, 29)( 31, 32)( 34, 35)( 37, 38)
( 39, 49)( 40, 48)( 41, 50)( 42, 52)( 43, 51)( 44, 53)( 45, 55)( 46, 54)
( 47, 56)( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)
( 64, 92)( 65, 91)( 66,103)( 67,102)( 68,104)( 69,106)( 70,105)( 71,107)
( 72,109)( 73,108)( 74,110)( 75, 94)( 76, 93)( 77, 95)( 78, 97)( 79, 96)
( 80, 98)( 81,100)( 82, 99)( 83,101);
s2 := Sym(110)!(  3, 66)(  4, 68)(  5, 67)(  6, 72)(  7, 74)(  8, 73)(  9, 69)
( 10, 71)( 11, 70)( 12, 57)( 13, 59)( 14, 58)( 15, 63)( 16, 65)( 17, 64)
( 18, 60)( 19, 62)( 20, 61)( 21, 76)( 22, 75)( 23, 77)( 24, 82)( 25, 81)
( 26, 83)( 27, 79)( 28, 78)( 29, 80)( 30, 93)( 31, 95)( 32, 94)( 33, 99)
( 34,101)( 35,100)( 36, 96)( 37, 98)( 38, 97)( 39, 84)( 40, 86)( 41, 85)
( 42, 90)( 43, 92)( 44, 91)( 45, 87)( 46, 89)( 47, 88)( 48,103)( 49,102)
( 50,104)( 51,109)( 52,108)( 53,110)( 54,106)( 55,105)( 56,107);
s3 := Sym(110)!(  3,  6)(  4,  7)(  5,  8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)
( 22, 25)( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)
( 48, 51)( 49, 52)( 50, 53)( 57, 60)( 58, 61)( 59, 62)( 66, 69)( 67, 70)
( 68, 71)( 75, 78)( 76, 79)( 77, 80)( 84, 87)( 85, 88)( 86, 89)( 93, 96)
( 94, 97)( 95, 98)(102,105)(103,106)(104,107);
poly := sub<Sym(110)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope