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Polytope of Type {2,36,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,36,4}*576c
if this polytope has a name.
Group : SmallGroup(576,4968)
Rank : 4
Schlafli Type : {2,36,4}
Number of vertices, edges, etc : 2, 36, 72, 4
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,36,4,2} of size 1152
Vertex Figure Of :
   {2,2,36,4} of size 1152
   {3,2,36,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,18,4}*288b
   3-fold quotients : {2,12,4}*192c
   4-fold quotients : {2,9,4}*144
   6-fold quotients : {2,6,4}*96c
   12-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,36,4}*1152c, {2,36,4}*1152b
   3-fold covers : {2,108,4}*1728c, {6,36,4}*1728d, {6,36,4}*1728f
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  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);;
s2 := (  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);;
s3 := (  3, 42)(  4, 41)(  5, 40)(  6, 39)(  7, 46)(  8, 45)(  9, 44)( 10, 43)
( 11, 50)( 12, 49)( 13, 48)( 14, 47)( 15, 54)( 16, 53)( 17, 52)( 18, 51)
( 19, 58)( 20, 57)( 21, 56)( 22, 55)( 23, 62)( 24, 61)( 25, 60)( 26, 59)
( 27, 66)( 28, 65)( 29, 64)( 30, 63)( 31, 70)( 32, 69)( 33, 68)( 34, 67)
( 35, 74)( 36, 73)( 37, 72)( 38, 71)( 75,114)( 76,113)( 77,112)( 78,111)
( 79,118)( 80,117)( 81,116)( 82,115)( 83,122)( 84,121)( 85,120)( 86,119)
( 87,126)( 88,125)( 89,124)( 90,123)( 91,130)( 92,129)( 93,128)( 94,127)
( 95,134)( 96,133)( 97,132)( 98,131)( 99,138)(100,137)(101,136)(102,135)
(103,142)(104,141)(105,140)(106,139)(107,146)(108,145)(109,144)(110,143);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(146)!(1,2);
s1 := 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);
s2 := 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);
s3 := Sym(146)!(  3, 42)(  4, 41)(  5, 40)(  6, 39)(  7, 46)(  8, 45)(  9, 44)
( 10, 43)( 11, 50)( 12, 49)( 13, 48)( 14, 47)( 15, 54)( 16, 53)( 17, 52)
( 18, 51)( 19, 58)( 20, 57)( 21, 56)( 22, 55)( 23, 62)( 24, 61)( 25, 60)
( 26, 59)( 27, 66)( 28, 65)( 29, 64)( 30, 63)( 31, 70)( 32, 69)( 33, 68)
( 34, 67)( 35, 74)( 36, 73)( 37, 72)( 38, 71)( 75,114)( 76,113)( 77,112)
( 78,111)( 79,118)( 80,117)( 81,116)( 82,115)( 83,122)( 84,121)( 85,120)
( 86,119)( 87,126)( 88,125)( 89,124)( 90,123)( 91,130)( 92,129)( 93,128)
( 94,127)( 95,134)( 96,133)( 97,132)( 98,131)( 99,138)(100,137)(101,136)
(102,135)(103,142)(104,141)(105,140)(106,139)(107,146)(108,145)(109,144)
(110,143);
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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