Polytope of Type {72,4}

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

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

Permutation Representation (Magma) :

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

References : None.
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