Polytope of Type {2,72,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,72,4}*1152a
if this polytope has a name.
Group : SmallGroup(1152,97526)
Rank : 4
Schlafli Type : {2,72,4}
Number of vertices, edges, etc : 2, 72, 144, 4
Order of s₀s₁s₂s₃ : 72
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,36,4}*576a, {2,72,2}*576
   3-fold quotients : {2,24,4}*384a
   4-fold quotients : {2,36,2}*288, {2,18,4}*288a
   6-fold quotients : {2,12,4}*192a, {2,24,2}*192
   8-fold quotients : {2,18,2}*144
   9-fold quotients : {2,8,4}*128a
   12-fold quotients : {2,12,2}*96, {2,6,4}*96a
   16-fold quotients : {2,9,2}*72
   18-fold quotients : {2,4,4}*64, {2,8,2}*64
   24-fold quotients : {2,6,2}*48
   36-fold quotients : {2,2,4}*32, {2,4,2}*32
   48-fold quotients : {2,3,2}*24
   72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

to this polytope