Polytope of Type {36,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,8}*576a
Also Known As : {36,8|2}. if this polytope has another name.
Group : SmallGroup(576,335)
Rank : 3
Schlafli Type : {36,8}
Number of vertices, edges, etc : 36, 144, 8
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {36,8,2} of size 1152
Vertex Figure Of :
   {2,36,8} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,4}*288a, {18,8}*288
   3-fold quotients : {12,8}*192a
   4-fold quotients : {36,2}*144, {18,4}*144a
   6-fold quotients : {12,4}*96a, {6,8}*96
   8-fold quotients : {18,2}*72
   9-fold quotients : {4,8}*64a
   12-fold quotients : {12,2}*48, {6,4}*48a
   16-fold quotients : {9,2}*36
   18-fold quotients : {4,4}*32, {2,8}*32
   24-fold quotients : {6,2}*24
   36-fold quotients : {2,4}*16, {4,2}*16
   48-fold quotients : {3,2}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,8}*1152a, {72,8}*1152a, {72,8}*1152c, {36,16}*1152a, {36,16}*1152b
   3-fold covers : {108,8}*1728a, {36,24}*1728b, {36,24}*1728c
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)( 15, 18)
( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)( 33, 36)
( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 47, 48)( 49, 53)( 50, 52)( 51, 54)
( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 65, 66)( 67, 71)( 68, 70)( 69, 72)
( 73,109)( 74,111)( 75,110)( 76,116)( 77,115)( 78,117)( 79,113)( 80,112)
( 81,114)( 82,118)( 83,120)( 84,119)( 85,125)( 86,124)( 87,126)( 88,122)
( 89,121)( 90,123)( 91,127)( 92,129)( 93,128)( 94,134)( 95,133)( 96,135)
( 97,131)( 98,130)( 99,132)(100,136)(101,138)(102,137)(103,143)(104,142)
(105,144)(106,140)(107,139)(108,141);;
s1 := (  1, 76)(  2, 78)(  3, 77)(  4, 73)(  5, 75)(  6, 74)(  7, 80)(  8, 79)
(  9, 81)( 10, 85)( 11, 87)( 12, 86)( 13, 82)( 14, 84)( 15, 83)( 16, 89)
( 17, 88)( 18, 90)( 19,103)( 20,105)( 21,104)( 22,100)( 23,102)( 24,101)
( 25,107)( 26,106)( 27,108)( 28, 94)( 29, 96)( 30, 95)( 31, 91)( 32, 93)
( 33, 92)( 34, 98)( 35, 97)( 36, 99)( 37,112)( 38,114)( 39,113)( 40,109)
( 41,111)( 42,110)( 43,116)( 44,115)( 45,117)( 46,121)( 47,123)( 48,122)
( 49,118)( 50,120)( 51,119)( 52,125)( 53,124)( 54,126)( 55,139)( 56,141)
( 57,140)( 58,136)( 59,138)( 60,137)( 61,143)( 62,142)( 63,144)( 64,130)
( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,134)( 71,133)( 72,135);;
s2 := ( 19, 28)( 20, 29)( 21, 30)( 22, 31)( 23, 32)( 24, 33)( 25, 34)( 26, 35)
( 27, 36)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)( 61, 70)
( 62, 71)( 63, 72)( 73, 91)( 74, 92)( 75, 93)( 76, 94)( 77, 95)( 78, 96)
( 79, 97)( 80, 98)( 81, 99)( 82,100)( 83,101)( 84,102)( 85,103)( 86,104)
( 87,105)( 88,106)( 89,107)( 90,108)(109,127)(110,128)(111,129)(112,130)
(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)
(121,139)(122,140)(123,141)(124,142)(125,143)(126,144);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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(144)!(  2,  3)(  4,  8)(  5,  7)(  6,  9)( 11, 12)( 13, 17)( 14, 16)
( 15, 18)( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)
( 33, 36)( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 47, 48)( 49, 53)( 50, 52)
( 51, 54)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 65, 66)( 67, 71)( 68, 70)
( 69, 72)( 73,109)( 74,111)( 75,110)( 76,116)( 77,115)( 78,117)( 79,113)
( 80,112)( 81,114)( 82,118)( 83,120)( 84,119)( 85,125)( 86,124)( 87,126)
( 88,122)( 89,121)( 90,123)( 91,127)( 92,129)( 93,128)( 94,134)( 95,133)
( 96,135)( 97,131)( 98,130)( 99,132)(100,136)(101,138)(102,137)(103,143)
(104,142)(105,144)(106,140)(107,139)(108,141);
s1 := Sym(144)!(  1, 76)(  2, 78)(  3, 77)(  4, 73)(  5, 75)(  6, 74)(  7, 80)
(  8, 79)(  9, 81)( 10, 85)( 11, 87)( 12, 86)( 13, 82)( 14, 84)( 15, 83)
( 16, 89)( 17, 88)( 18, 90)( 19,103)( 20,105)( 21,104)( 22,100)( 23,102)
( 24,101)( 25,107)( 26,106)( 27,108)( 28, 94)( 29, 96)( 30, 95)( 31, 91)
( 32, 93)( 33, 92)( 34, 98)( 35, 97)( 36, 99)( 37,112)( 38,114)( 39,113)
( 40,109)( 41,111)( 42,110)( 43,116)( 44,115)( 45,117)( 46,121)( 47,123)
( 48,122)( 49,118)( 50,120)( 51,119)( 52,125)( 53,124)( 54,126)( 55,139)
( 56,141)( 57,140)( 58,136)( 59,138)( 60,137)( 61,143)( 62,142)( 63,144)
( 64,130)( 65,132)( 66,131)( 67,127)( 68,129)( 69,128)( 70,134)( 71,133)
( 72,135);
s2 := Sym(144)!( 19, 28)( 20, 29)( 21, 30)( 22, 31)( 23, 32)( 24, 33)( 25, 34)
( 26, 35)( 27, 36)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 73, 91)( 74, 92)( 75, 93)( 76, 94)( 77, 95)
( 78, 96)( 79, 97)( 80, 98)( 81, 99)( 82,100)( 83,101)( 84,102)( 85,103)
( 86,104)( 87,105)( 88,106)( 89,107)( 90,108)(109,127)(110,128)(111,129)
(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)
(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144);
poly := sub<Sym(144)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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.
to this polytope