Polytope of Type {4,72}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,72}*1152b
if this polytope has a name.
Group : SmallGroup(1152,32532)
Rank : 3
Schlafli Type : {4,72}
Number of vertices, edges, etc : 8, 288, 144
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,36}*576a
   3-fold quotients : {4,24}*384b
   4-fold quotients : {4,36}*288a
   6-fold quotients : {4,12}*192a
   8-fold quotients : {2,36}*144, {4,18}*144a
   9-fold quotients : {4,8}*128b
   12-fold quotients : {4,12}*96a
   16-fold quotients : {2,18}*72
   18-fold quotients : {4,4}*64
   24-fold quotients : {2,12}*48, {4,6}*48a
   32-fold quotients : {2,9}*36
   36-fold quotients : {4,4}*32
   48-fold quotients : {2,6}*24
   72-fold quotients : {2,4}*16, {4,2}*16
   96-fold quotients : {2,3}*12
   144-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope