Polytope of Type {2,4,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,36}*576c
if this polytope has a name.
Group : SmallGroup(576,4968)
Rank : 4
Schlafli Type : {2,4,36}
Number of vertices, edges, etc : 2, 4, 72, 36
Order of s₀s₁s₂s₃ : 36
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,36,2} of size 1152
Vertex Figure Of :
   {2,2,4,36} of size 1152
   {3,2,4,36} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,18}*288b
   3-fold quotients : {2,4,12}*192c
   4-fold quotients : {2,4,9}*144
   6-fold quotients : {2,4,6}*96c
   12-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,4,36}*1152b
   3-fold covers : {2,4,108}*1728c
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  3, 40)(  4, 39)(  5, 42)(  6, 41)(  7, 44)(  8, 43)(  9, 46)( 10, 45)
( 11, 48)( 12, 47)( 13, 50)( 14, 49)( 15, 52)( 16, 51)( 17, 54)( 18, 53)
( 19, 56)( 20, 55)( 21, 58)( 22, 57)( 23, 60)( 24, 59)( 25, 62)( 26, 61)
( 27, 64)( 28, 63)( 29, 66)( 30, 65)( 31, 68)( 32, 67)( 33, 70)( 34, 69)
( 35, 72)( 36, 71)( 37, 74)( 38, 73)( 75,112)( 76,111)( 77,114)( 78,113)
( 79,116)( 80,115)( 81,118)( 82,117)( 83,120)( 84,119)( 85,122)( 86,121)
( 87,124)( 88,123)( 89,126)( 90,125)( 91,128)( 92,127)( 93,130)( 94,129)
( 95,132)( 96,131)( 97,134)( 98,133)( 99,136)(100,135)(101,138)(102,137)
(103,140)(104,139)(105,142)(106,141)(107,144)(108,143)(109,146)(110,145);;
s2 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 31)( 16, 33)( 17, 32)
( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)( 25, 36)
( 26, 38)( 40, 41)( 43, 47)( 44, 49)( 45, 48)( 46, 50)( 51, 67)( 52, 69)
( 53, 68)( 54, 70)( 55, 63)( 56, 65)( 57, 64)( 58, 66)( 59, 71)( 60, 73)
( 61, 72)( 62, 74)( 75,111)( 76,113)( 77,112)( 78,114)( 79,119)( 80,121)
( 81,120)( 82,122)( 83,115)( 84,117)( 85,116)( 86,118)( 87,139)( 88,141)
( 89,140)( 90,142)( 91,135)( 92,137)( 93,136)( 94,138)( 95,143)( 96,145)
( 97,144)( 98,146)( 99,127)(100,129)(101,128)(102,130)(103,123)(104,125)
(105,124)(106,126)(107,131)(108,133)(109,132)(110,134);;
s3 := (  3, 87)(  4, 88)(  5, 90)(  6, 89)(  7, 95)(  8, 96)(  9, 98)( 10, 97)
( 11, 91)( 12, 92)( 13, 94)( 14, 93)( 15, 75)( 16, 76)( 17, 78)( 18, 77)
( 19, 83)( 20, 84)( 21, 86)( 22, 85)( 23, 79)( 24, 80)( 25, 82)( 26, 81)
( 27,103)( 28,104)( 29,106)( 30,105)( 31, 99)( 32,100)( 33,102)( 34,101)
( 35,107)( 36,108)( 37,110)( 38,109)( 39,123)( 40,124)( 41,126)( 42,125)
( 43,131)( 44,132)( 45,134)( 46,133)( 47,127)( 48,128)( 49,130)( 50,129)
( 51,111)( 52,112)( 53,114)( 54,113)( 55,119)( 56,120)( 57,122)( 58,121)
( 59,115)( 60,116)( 61,118)( 62,117)( 63,139)( 64,140)( 65,142)( 66,141)
( 67,135)( 68,136)( 69,138)( 70,137)( 71,143)( 72,144)( 73,146)( 74,145);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(146)!(1,2);
s1 := Sym(146)!(  3, 40)(  4, 39)(  5, 42)(  6, 41)(  7, 44)(  8, 43)(  9, 46)
( 10, 45)( 11, 48)( 12, 47)( 13, 50)( 14, 49)( 15, 52)( 16, 51)( 17, 54)
( 18, 53)( 19, 56)( 20, 55)( 21, 58)( 22, 57)( 23, 60)( 24, 59)( 25, 62)
( 26, 61)( 27, 64)( 28, 63)( 29, 66)( 30, 65)( 31, 68)( 32, 67)( 33, 70)
( 34, 69)( 35, 72)( 36, 71)( 37, 74)( 38, 73)( 75,112)( 76,111)( 77,114)
( 78,113)( 79,116)( 80,115)( 81,118)( 82,117)( 83,120)( 84,119)( 85,122)
( 86,121)( 87,124)( 88,123)( 89,126)( 90,125)( 91,128)( 92,127)( 93,130)
( 94,129)( 95,132)( 96,131)( 97,134)( 98,133)( 99,136)(100,135)(101,138)
(102,137)(103,140)(104,139)(105,142)(106,141)(107,144)(108,143)(109,146)
(110,145);
s2 := Sym(146)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 31)( 16, 33)
( 17, 32)( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)
( 25, 36)( 26, 38)( 40, 41)( 43, 47)( 44, 49)( 45, 48)( 46, 50)( 51, 67)
( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)( 57, 64)( 58, 66)( 59, 71)
( 60, 73)( 61, 72)( 62, 74)( 75,111)( 76,113)( 77,112)( 78,114)( 79,119)
( 80,121)( 81,120)( 82,122)( 83,115)( 84,117)( 85,116)( 86,118)( 87,139)
( 88,141)( 89,140)( 90,142)( 91,135)( 92,137)( 93,136)( 94,138)( 95,143)
( 96,145)( 97,144)( 98,146)( 99,127)(100,129)(101,128)(102,130)(103,123)
(104,125)(105,124)(106,126)(107,131)(108,133)(109,132)(110,134);
s3 := Sym(146)!(  3, 87)(  4, 88)(  5, 90)(  6, 89)(  7, 95)(  8, 96)(  9, 98)
( 10, 97)( 11, 91)( 12, 92)( 13, 94)( 14, 93)( 15, 75)( 16, 76)( 17, 78)
( 18, 77)( 19, 83)( 20, 84)( 21, 86)( 22, 85)( 23, 79)( 24, 80)( 25, 82)
( 26, 81)( 27,103)( 28,104)( 29,106)( 30,105)( 31, 99)( 32,100)( 33,102)
( 34,101)( 35,107)( 36,108)( 37,110)( 38,109)( 39,123)( 40,124)( 41,126)
( 42,125)( 43,131)( 44,132)( 45,134)( 46,133)( 47,127)( 48,128)( 49,130)
( 50,129)( 51,111)( 52,112)( 53,114)( 54,113)( 55,119)( 56,120)( 57,122)
( 58,121)( 59,115)( 60,116)( 61,118)( 62,117)( 63,139)( 64,140)( 65,142)
( 66,141)( 67,135)( 68,136)( 69,138)( 70,137)( 71,143)( 72,144)( 73,146)
( 74,145);
poly := sub<Sym(146)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s3 >;

to this polytope