Polytope of Type {4,18,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,10}*1440a
Also Known As : {{4,18|2},{18,10|2}}. if this polytope has another name.
Group : SmallGroup(1440,1593)
Rank : 4
Schlafli Type : {4,18,10}
Number of vertices, edges, etc : 4, 36, 90, 10
Order of s₀s₁s₂s₃ : 180
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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 : {2,18,10}*720
   3-fold quotients : {4,6,10}*480a
   5-fold quotients : {4,18,2}*288a
   6-fold quotients : {2,6,10}*240
   9-fold quotients : {4,2,10}*160
   10-fold quotients : {2,18,2}*144
   15-fold quotients : {4,6,2}*96a
   18-fold quotients : {4,2,5}*80, {2,2,10}*80
   20-fold quotients : {2,9,2}*72
   30-fold quotients : {2,6,2}*48
   36-fold quotients : {2,2,5}*40
   45-fold quotients : {4,2,2}*32
   60-fold quotients : {2,3,2}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)
( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,151)
(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)(114,159)
(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)(122,167)
(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)(130,175)
(131,176)(132,177)(133,178)(134,179)(135,180);;
s1 := (  1, 91)(  2, 93)(  3, 92)(  4, 94)(  5, 96)(  6, 95)(  7, 97)(  8, 99)
(  9, 98)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)( 16,123)
( 17,122)( 18,121)( 19,126)( 20,125)( 21,124)( 22,129)( 23,128)( 24,127)
( 25,132)( 26,131)( 27,130)( 28,135)( 29,134)( 30,133)( 31,108)( 32,107)
( 33,106)( 34,111)( 35,110)( 36,109)( 37,114)( 38,113)( 39,112)( 40,117)
( 41,116)( 42,115)( 43,120)( 44,119)( 45,118)( 46,136)( 47,138)( 48,137)
( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)( 56,147)
( 57,146)( 58,148)( 59,150)( 60,149)( 61,168)( 62,167)( 63,166)( 64,171)
( 65,170)( 66,169)( 67,174)( 68,173)( 69,172)( 70,177)( 71,176)( 72,175)
( 73,180)( 74,179)( 75,178)( 76,153)( 77,152)( 78,151)( 79,156)( 80,155)
( 81,154)( 82,159)( 83,158)( 84,157)( 85,162)( 86,161)( 87,160)( 88,165)
( 89,164)( 90,163);;
s2 := (  1, 16)(  2, 18)(  3, 17)(  4, 28)(  5, 30)(  6, 29)(  7, 25)(  8, 27)
(  9, 26)( 10, 22)( 11, 24)( 12, 23)( 13, 19)( 14, 21)( 15, 20)( 31, 33)
( 34, 45)( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 46, 61)( 47, 63)
( 48, 62)( 49, 73)( 50, 75)( 51, 74)( 52, 70)( 53, 72)( 54, 71)( 55, 67)
( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 76, 78)( 79, 90)( 80, 89)
( 81, 88)( 82, 87)( 83, 86)( 84, 85)( 91,106)( 92,108)( 93,107)( 94,118)
( 95,120)( 96,119)( 97,115)( 98,117)( 99,116)(100,112)(101,114)(102,113)
(103,109)(104,111)(105,110)(121,123)(124,135)(125,134)(126,133)(127,132)
(128,131)(129,130)(136,151)(137,153)(138,152)(139,163)(140,165)(141,164)
(142,160)(143,162)(144,161)(145,157)(146,159)(147,158)(148,154)(149,156)
(150,155)(166,168)(169,180)(170,179)(171,178)(172,177)(173,176)(174,175);;
s3 := (  1,  4)(  2,  5)(  3,  6)(  7, 13)(  8, 14)(  9, 15)( 16, 19)( 17, 20)
( 18, 21)( 22, 28)( 23, 29)( 24, 30)( 31, 34)( 32, 35)( 33, 36)( 37, 43)
( 38, 44)( 39, 45)( 46, 49)( 47, 50)( 48, 51)( 52, 58)( 53, 59)( 54, 60)
( 61, 64)( 62, 65)( 63, 66)( 67, 73)( 68, 74)( 69, 75)( 76, 79)( 77, 80)
( 78, 81)( 82, 88)( 83, 89)( 84, 90)( 91, 94)( 92, 95)( 93, 96)( 97,103)
( 98,104)( 99,105)(106,109)(107,110)(108,111)(112,118)(113,119)(114,120)
(121,124)(122,125)(123,126)(127,133)(128,134)(129,135)(136,139)(137,140)
(138,141)(142,148)(143,149)(144,150)(151,154)(152,155)(153,156)(157,163)
(158,164)(159,165)(166,169)(167,170)(168,171)(172,178)(173,179)(174,180);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(180)!( 91,136)( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)
( 98,143)( 99,144)(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)
(106,151)(107,152)(108,153)(109,154)(110,155)(111,156)(112,157)(113,158)
(114,159)(115,160)(116,161)(117,162)(118,163)(119,164)(120,165)(121,166)
(122,167)(123,168)(124,169)(125,170)(126,171)(127,172)(128,173)(129,174)
(130,175)(131,176)(132,177)(133,178)(134,179)(135,180);
s1 := Sym(180)!(  1, 91)(  2, 93)(  3, 92)(  4, 94)(  5, 96)(  6, 95)(  7, 97)
(  8, 99)(  9, 98)( 10,100)( 11,102)( 12,101)( 13,103)( 14,105)( 15,104)
( 16,123)( 17,122)( 18,121)( 19,126)( 20,125)( 21,124)( 22,129)( 23,128)
( 24,127)( 25,132)( 26,131)( 27,130)( 28,135)( 29,134)( 30,133)( 31,108)
( 32,107)( 33,106)( 34,111)( 35,110)( 36,109)( 37,114)( 38,113)( 39,112)
( 40,117)( 41,116)( 42,115)( 43,120)( 44,119)( 45,118)( 46,136)( 47,138)
( 48,137)( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)
( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,168)( 62,167)( 63,166)
( 64,171)( 65,170)( 66,169)( 67,174)( 68,173)( 69,172)( 70,177)( 71,176)
( 72,175)( 73,180)( 74,179)( 75,178)( 76,153)( 77,152)( 78,151)( 79,156)
( 80,155)( 81,154)( 82,159)( 83,158)( 84,157)( 85,162)( 86,161)( 87,160)
( 88,165)( 89,164)( 90,163);
s2 := Sym(180)!(  1, 16)(  2, 18)(  3, 17)(  4, 28)(  5, 30)(  6, 29)(  7, 25)
(  8, 27)(  9, 26)( 10, 22)( 11, 24)( 12, 23)( 13, 19)( 14, 21)( 15, 20)
( 31, 33)( 34, 45)( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 46, 61)
( 47, 63)( 48, 62)( 49, 73)( 50, 75)( 51, 74)( 52, 70)( 53, 72)( 54, 71)
( 55, 67)( 56, 69)( 57, 68)( 58, 64)( 59, 66)( 60, 65)( 76, 78)( 79, 90)
( 80, 89)( 81, 88)( 82, 87)( 83, 86)( 84, 85)( 91,106)( 92,108)( 93,107)
( 94,118)( 95,120)( 96,119)( 97,115)( 98,117)( 99,116)(100,112)(101,114)
(102,113)(103,109)(104,111)(105,110)(121,123)(124,135)(125,134)(126,133)
(127,132)(128,131)(129,130)(136,151)(137,153)(138,152)(139,163)(140,165)
(141,164)(142,160)(143,162)(144,161)(145,157)(146,159)(147,158)(148,154)
(149,156)(150,155)(166,168)(169,180)(170,179)(171,178)(172,177)(173,176)
(174,175);
s3 := Sym(180)!(  1,  4)(  2,  5)(  3,  6)(  7, 13)(  8, 14)(  9, 15)( 16, 19)
( 17, 20)( 18, 21)( 22, 28)( 23, 29)( 24, 30)( 31, 34)( 32, 35)( 33, 36)
( 37, 43)( 38, 44)( 39, 45)( 46, 49)( 47, 50)( 48, 51)( 52, 58)( 53, 59)
( 54, 60)( 61, 64)( 62, 65)( 63, 66)( 67, 73)( 68, 74)( 69, 75)( 76, 79)
( 77, 80)( 78, 81)( 82, 88)( 83, 89)( 84, 90)( 91, 94)( 92, 95)( 93, 96)
( 97,103)( 98,104)( 99,105)(106,109)(107,110)(108,111)(112,118)(113,119)
(114,120)(121,124)(122,125)(123,126)(127,133)(128,134)(129,135)(136,139)
(137,140)(138,141)(142,148)(143,149)(144,150)(151,154)(152,155)(153,156)
(157,163)(158,164)(159,165)(166,169)(167,170)(168,171)(172,178)(173,179)
(174,180);
poly := sub<Sym(180)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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 >;

References : None.
to this polytope