Polytope of Type {72,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,12249)
Rank : 3
Schlafli Type : {72,6}
Number of vertices, edges, etc : 144, 432, 12
Order of s₀s₁s₂ : 9
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {24,6}*576a
   4-fold quotients : {36,6}*432c
   9-fold quotients : {8,6}*192a
   12-fold quotients : {12,6}*144d
   36-fold quotients : {4,6}*48b
   72-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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