Polytope of Type {45,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {45,4}*360
if this polytope has a name.
Group : SmallGroup(360,41)
Rank : 3
Schlafli Type : {45,4}
Number of vertices, edges, etc : 45, 90, 4
Order of s₀s₁s₂ : 45
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {45,4,2} of size 720
Vertex Figure Of :
   {2,45,4} of size 720
   {4,45,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {15,4}*120
   5-fold quotients : {9,4}*72
   15-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {45,4}*720, {90,4}*720b, {90,4}*720c
   3-fold covers : {135,4}*1080
   4-fold covers : {180,4}*1440b, {180,4}*1440c, {45,8}*1440, {90,4}*1440
   5-fold covers : {225,4}*1800
Permutation Representation (GAP) :

s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 49)( 14, 51)( 15, 50)
( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)( 23, 54)
( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)( 31, 46)
( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)( 63,126)
( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)( 71,130)
( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)( 79,170)
( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)( 87,162)
( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)( 95,166)
( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)(103,146)
(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)(111,138)
(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)(119,142)
(120,144);;
s1 := (  1, 73)(  2, 74)(  3, 76)(  4, 75)(  5, 81)(  6, 82)(  7, 84)(  8, 83)
(  9, 77)( 10, 78)( 11, 80)( 12, 79)( 13, 61)( 14, 62)( 15, 64)( 16, 63)
( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)( 24, 67)
( 25,109)( 26,110)( 27,112)( 28,111)( 29,117)( 30,118)( 31,120)( 32,119)
( 33,113)( 34,114)( 35,116)( 36,115)( 37, 97)( 38, 98)( 39,100)( 40, 99)
( 41,105)( 42,106)( 43,108)( 44,107)( 45,101)( 46,102)( 47,104)( 48,103)
( 49, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 93)( 54, 94)( 55, 96)( 56, 95)
( 57, 89)( 58, 90)( 59, 92)( 60, 91)(121,137)(122,138)(123,140)(124,139)
(125,133)(126,134)(127,136)(128,135)(129,141)(130,142)(131,144)(132,143)
(145,173)(146,174)(147,176)(148,175)(149,169)(150,170)(151,172)(152,171)
(153,177)(154,178)(155,180)(156,179)(157,161)(158,162)(159,164)(160,163)
(167,168);;
s2 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*s2*s1*s2*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(180)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 49)( 14, 51)
( 15, 50)( 16, 52)( 17, 57)( 18, 59)( 19, 58)( 20, 60)( 21, 53)( 22, 55)
( 23, 54)( 24, 56)( 25, 37)( 26, 39)( 27, 38)( 28, 40)( 29, 45)( 30, 47)
( 31, 46)( 32, 48)( 33, 41)( 34, 43)( 35, 42)( 36, 44)( 61,125)( 62,127)
( 63,126)( 64,128)( 65,121)( 66,123)( 67,122)( 68,124)( 69,129)( 70,131)
( 71,130)( 72,132)( 73,173)( 74,175)( 75,174)( 76,176)( 77,169)( 78,171)
( 79,170)( 80,172)( 81,177)( 82,179)( 83,178)( 84,180)( 85,161)( 86,163)
( 87,162)( 88,164)( 89,157)( 90,159)( 91,158)( 92,160)( 93,165)( 94,167)
( 95,166)( 96,168)( 97,149)( 98,151)( 99,150)(100,152)(101,145)(102,147)
(103,146)(104,148)(105,153)(106,155)(107,154)(108,156)(109,137)(110,139)
(111,138)(112,140)(113,133)(114,135)(115,134)(116,136)(117,141)(118,143)
(119,142)(120,144);
s1 := Sym(180)!(  1, 73)(  2, 74)(  3, 76)(  4, 75)(  5, 81)(  6, 82)(  7, 84)
(  8, 83)(  9, 77)( 10, 78)( 11, 80)( 12, 79)( 13, 61)( 14, 62)( 15, 64)
( 16, 63)( 17, 69)( 18, 70)( 19, 72)( 20, 71)( 21, 65)( 22, 66)( 23, 68)
( 24, 67)( 25,109)( 26,110)( 27,112)( 28,111)( 29,117)( 30,118)( 31,120)
( 32,119)( 33,113)( 34,114)( 35,116)( 36,115)( 37, 97)( 38, 98)( 39,100)
( 40, 99)( 41,105)( 42,106)( 43,108)( 44,107)( 45,101)( 46,102)( 47,104)
( 48,103)( 49, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 93)( 54, 94)( 55, 96)
( 56, 95)( 57, 89)( 58, 90)( 59, 92)( 60, 91)(121,137)(122,138)(123,140)
(124,139)(125,133)(126,134)(127,136)(128,135)(129,141)(130,142)(131,144)
(132,143)(145,173)(146,174)(147,176)(148,175)(149,169)(150,170)(151,172)
(152,171)(153,177)(154,178)(155,180)(156,179)(157,161)(158,162)(159,164)
(160,163)(167,168);
s2 := Sym(180)!(  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)
(174,175)(177,180)(178,179);
poly := sub<Sym(180)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*s2*s1*s2*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