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Polytope of Type {4,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,36}*576c
if this polytope has a name.
Group : SmallGroup(576,4970)
Rank : 3
Schlafli Type : {4,36}
Number of vertices, edges, etc : 8, 144, 72
Order of s0s1s2 : 18
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,36,2} of size 1152
Vertex Figure Of :
   {2,4,36} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,18}*288
   3-fold quotients : {4,12}*192c
   4-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c
   6-fold quotients : {4,6}*96
   8-fold quotients : {4,9}*72, {2,18}*72
   12-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   16-fold quotients : {2,9}*36
   24-fold quotients : {4,3}*24, {2,6}*24
   48-fold quotients : {2,3}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,36}*1152d, {8,36}*1152g, {8,36}*1152h, {4,72}*1152e, {4,72}*1152f
   3-fold covers : {4,108}*1728c, {12,36}*1728g, {12,36}*1728h
Permutation Representation (GAP) :
s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73,111)( 74,112)( 75,109)( 76,110)
( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)( 84,118)
( 85,123)( 86,124)( 87,121)( 88,122)( 89,127)( 90,128)( 91,125)( 92,126)
( 93,131)( 94,132)( 95,129)( 96,130)( 97,135)( 98,136)( 99,133)(100,134)
(101,139)(102,140)(103,137)(104,138)(105,143)(106,144)(107,141)(108,142);;
s1 := (  1, 73)(  2, 75)(  3, 74)(  4, 76)(  5, 81)(  6, 83)(  7, 82)(  8, 84)
(  9, 77)( 10, 79)( 11, 78)( 12, 80)( 13,105)( 14,107)( 15,106)( 16,108)
( 17,101)( 18,103)( 19,102)( 20,104)( 21, 97)( 22, 99)( 23, 98)( 24,100)
( 25, 93)( 26, 95)( 27, 94)( 28, 96)( 29, 89)( 30, 91)( 31, 90)( 32, 92)
( 33, 85)( 34, 87)( 35, 86)( 36, 88)( 37,109)( 38,111)( 39,110)( 40,112)
( 41,117)( 42,119)( 43,118)( 44,120)( 45,113)( 46,115)( 47,114)( 48,116)
( 49,141)( 50,143)( 51,142)( 52,144)( 53,137)( 54,139)( 55,138)( 56,140)
( 57,133)( 58,135)( 59,134)( 60,136)( 61,129)( 62,131)( 63,130)( 64,132)
( 65,125)( 66,127)( 67,126)( 68,128)( 69,121)( 70,123)( 71,122)( 72,124);;
s2 := (  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 33)(  6, 36)(  7, 35)(  8, 34)
(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 14, 16)( 17, 21)( 18, 24)( 19, 23)
( 20, 22)( 37, 61)( 38, 64)( 39, 63)( 40, 62)( 41, 69)( 42, 72)( 43, 71)
( 44, 70)( 45, 65)( 46, 68)( 47, 67)( 48, 66)( 50, 52)( 53, 57)( 54, 60)
( 55, 59)( 56, 58)( 73,133)( 74,136)( 75,135)( 76,134)( 77,141)( 78,144)
( 79,143)( 80,142)( 81,137)( 82,140)( 83,139)( 84,138)( 85,121)( 86,124)
( 87,123)( 88,122)( 89,129)( 90,132)( 91,131)( 92,130)( 93,125)( 94,128)
( 95,127)( 96,126)( 97,109)( 98,112)( 99,111)(100,110)(101,117)(102,120)
(103,119)(104,118)(105,113)(106,116)(107,115)(108,114);;
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, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*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*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(144)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73,111)( 74,112)( 75,109)
( 76,110)( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)
( 84,118)( 85,123)( 86,124)( 87,121)( 88,122)( 89,127)( 90,128)( 91,125)
( 92,126)( 93,131)( 94,132)( 95,129)( 96,130)( 97,135)( 98,136)( 99,133)
(100,134)(101,139)(102,140)(103,137)(104,138)(105,143)(106,144)(107,141)
(108,142);
s1 := Sym(144)!(  1, 73)(  2, 75)(  3, 74)(  4, 76)(  5, 81)(  6, 83)(  7, 82)
(  8, 84)(  9, 77)( 10, 79)( 11, 78)( 12, 80)( 13,105)( 14,107)( 15,106)
( 16,108)( 17,101)( 18,103)( 19,102)( 20,104)( 21, 97)( 22, 99)( 23, 98)
( 24,100)( 25, 93)( 26, 95)( 27, 94)( 28, 96)( 29, 89)( 30, 91)( 31, 90)
( 32, 92)( 33, 85)( 34, 87)( 35, 86)( 36, 88)( 37,109)( 38,111)( 39,110)
( 40,112)( 41,117)( 42,119)( 43,118)( 44,120)( 45,113)( 46,115)( 47,114)
( 48,116)( 49,141)( 50,143)( 51,142)( 52,144)( 53,137)( 54,139)( 55,138)
( 56,140)( 57,133)( 58,135)( 59,134)( 60,136)( 61,129)( 62,131)( 63,130)
( 64,132)( 65,125)( 66,127)( 67,126)( 68,128)( 69,121)( 70,123)( 71,122)
( 72,124);
s2 := Sym(144)!(  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 33)(  6, 36)(  7, 35)
(  8, 34)(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 14, 16)( 17, 21)( 18, 24)
( 19, 23)( 20, 22)( 37, 61)( 38, 64)( 39, 63)( 40, 62)( 41, 69)( 42, 72)
( 43, 71)( 44, 70)( 45, 65)( 46, 68)( 47, 67)( 48, 66)( 50, 52)( 53, 57)
( 54, 60)( 55, 59)( 56, 58)( 73,133)( 74,136)( 75,135)( 76,134)( 77,141)
( 78,144)( 79,143)( 80,142)( 81,137)( 82,140)( 83,139)( 84,138)( 85,121)
( 86,124)( 87,123)( 88,122)( 89,129)( 90,132)( 91,131)( 92,130)( 93,125)
( 94,128)( 95,127)( 96,126)( 97,109)( 98,112)( 99,111)(100,110)(101,117)
(102,120)(103,119)(104,118)(105,113)(106,116)(107,115)(108,114);
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, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*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*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1 >; 
 
References : None.
to this polytope