Polytope of Type {72,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,4}*576a
Also Known As : {72,4|2}. if this polytope has another name.
Group : SmallGroup(576,296)
Rank : 3
Schlafli Type : {72,4}
Number of vertices, edges, etc : 72, 144, 4
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {72,4,2} of size 1152
Vertex Figure Of :
   {2,72,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,4}*288a, {72,2}*288
   3-fold quotients : {24,4}*192a
   4-fold quotients : {36,2}*144, {18,4}*144a
   6-fold quotients : {12,4}*96a, {24,2}*96
   8-fold quotients : {18,2}*72
   9-fold quotients : {8,4}*64a
   12-fold quotients : {12,2}*48, {6,4}*48a
   16-fold quotients : {9,2}*36
   18-fold quotients : {4,4}*32, {8,2}*32
   24-fold quotients : {6,2}*24
   36-fold quotients : {2,4}*16, {4,2}*16
   48-fold quotients : {3,2}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {72,4}*1152a, {72,8}*1152b, {72,8}*1152c, {144,4}*1152a, {144,4}*1152b
   3-fold covers : {216,4}*1728a, {72,12}*1728a, {72,12}*1728b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope