Polytope of Type {8,36}

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

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

References : None.
to this polytope