Polytope of Type {2,4,72}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,72}*1152b
if this polytope has a name.
Group : SmallGroup(1152,98780)
Rank : 4
Schlafli Type : {2,4,72}
Number of vertices, edges, etc : 2, 4, 144, 72
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 : {2,4,36}*576a
   3-fold quotients : {2,4,24}*384b
   4-fold quotients : {2,2,36}*288, {2,4,18}*288a
   6-fold quotients : {2,4,12}*192a
   8-fold quotients : {2,2,18}*144
   9-fold quotients : {2,4,8}*128b
   12-fold quotients : {2,2,12}*96, {2,4,6}*96a
   16-fold quotients : {2,2,9}*72
   18-fold quotients : {2,4,4}*64
   24-fold quotients : {2,2,6}*48
   36-fold quotients : {2,2,4}*32, {2,4,2}*32
   48-fold quotients : {2,2,3}*24
   72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope