Polytope of Type {2,36,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,36,10}*1440
if this polytope has a name.
Group : SmallGroup(1440,1583)
Rank : 4
Schlafli Type : {2,36,10}
Number of vertices, edges, etc : 2, 36, 180, 10
Order of s₀s₁s₂s₃ : 180
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 : {2,18,10}*720
   3-fold quotients : {2,12,10}*480
   5-fold quotients : {2,36,2}*288
   6-fold quotients : {2,6,10}*240
   9-fold quotients : {2,4,10}*160
   10-fold quotients : {2,18,2}*144
   15-fold quotients : {2,12,2}*96
   18-fold quotients : {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 : {2,4,2}*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 := (1,2);;
s1 := (  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 18, 35)( 19, 34)( 20, 33)
( 21, 38)( 22, 37)( 23, 36)( 24, 41)( 25, 40)( 26, 39)( 27, 44)( 28, 43)
( 29, 42)( 30, 47)( 31, 46)( 32, 45)( 49, 50)( 52, 53)( 55, 56)( 58, 59)
( 61, 62)( 63, 80)( 64, 79)( 65, 78)( 66, 83)( 67, 82)( 68, 81)( 69, 86)
( 70, 85)( 71, 84)( 72, 89)( 73, 88)( 74, 87)( 75, 92)( 76, 91)( 77, 90)
( 93,138)( 94,140)( 95,139)( 96,141)( 97,143)( 98,142)( 99,144)(100,146)
(101,145)(102,147)(103,149)(104,148)(105,150)(106,152)(107,151)(108,170)
(109,169)(110,168)(111,173)(112,172)(113,171)(114,176)(115,175)(116,174)
(117,179)(118,178)(119,177)(120,182)(121,181)(122,180)(123,155)(124,154)
(125,153)(126,158)(127,157)(128,156)(129,161)(130,160)(131,159)(132,164)
(133,163)(134,162)(135,167)(136,166)(137,165);;
s2 := (  3,108)(  4,110)(  5,109)(  6,120)(  7,122)(  8,121)(  9,117)( 10,119)
( 11,118)( 12,114)( 13,116)( 14,115)( 15,111)( 16,113)( 17,112)( 18, 93)
( 19, 95)( 20, 94)( 21,105)( 22,107)( 23,106)( 24,102)( 25,104)( 26,103)
( 27, 99)( 28,101)( 29,100)( 30, 96)( 31, 98)( 32, 97)( 33,125)( 34,124)
( 35,123)( 36,137)( 37,136)( 38,135)( 39,134)( 40,133)( 41,132)( 42,131)
( 43,130)( 44,129)( 45,128)( 46,127)( 47,126)( 48,153)( 49,155)( 50,154)
( 51,165)( 52,167)( 53,166)( 54,162)( 55,164)( 56,163)( 57,159)( 58,161)
( 59,160)( 60,156)( 61,158)( 62,157)( 63,138)( 64,140)( 65,139)( 66,150)
( 67,152)( 68,151)( 69,147)( 70,149)( 71,148)( 72,144)( 73,146)( 74,145)
( 75,141)( 76,143)( 77,142)( 78,170)( 79,169)( 80,168)( 81,182)( 82,181)
( 83,180)( 84,179)( 85,178)( 86,177)( 87,176)( 88,175)( 89,174)( 90,173)
( 91,172)( 92,171);;
s3 := (  3,  6)(  4,  7)(  5,  8)(  9, 15)( 10, 16)( 11, 17)( 18, 21)( 19, 22)
( 20, 23)( 24, 30)( 25, 31)( 26, 32)( 33, 36)( 34, 37)( 35, 38)( 39, 45)
( 40, 46)( 41, 47)( 48, 51)( 49, 52)( 50, 53)( 54, 60)( 55, 61)( 56, 62)
( 63, 66)( 64, 67)( 65, 68)( 69, 75)( 70, 76)( 71, 77)( 78, 81)( 79, 82)
( 80, 83)( 84, 90)( 85, 91)( 86, 92)( 93, 96)( 94, 97)( 95, 98)( 99,105)
(100,106)(101,107)(108,111)(109,112)(110,113)(114,120)(115,121)(116,122)
(123,126)(124,127)(125,128)(129,135)(130,136)(131,137)(138,141)(139,142)
(140,143)(144,150)(145,151)(146,152)(153,156)(154,157)(155,158)(159,165)
(160,166)(161,167)(168,171)(169,172)(170,173)(174,180)(175,181)(176,182);;
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*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*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(182)!(1,2);
s1 := Sym(182)!(  4,  5)(  7,  8)( 10, 11)( 13, 14)( 16, 17)( 18, 35)( 19, 34)
( 20, 33)( 21, 38)( 22, 37)( 23, 36)( 24, 41)( 25, 40)( 26, 39)( 27, 44)
( 28, 43)( 29, 42)( 30, 47)( 31, 46)( 32, 45)( 49, 50)( 52, 53)( 55, 56)
( 58, 59)( 61, 62)( 63, 80)( 64, 79)( 65, 78)( 66, 83)( 67, 82)( 68, 81)
( 69, 86)( 70, 85)( 71, 84)( 72, 89)( 73, 88)( 74, 87)( 75, 92)( 76, 91)
( 77, 90)( 93,138)( 94,140)( 95,139)( 96,141)( 97,143)( 98,142)( 99,144)
(100,146)(101,145)(102,147)(103,149)(104,148)(105,150)(106,152)(107,151)
(108,170)(109,169)(110,168)(111,173)(112,172)(113,171)(114,176)(115,175)
(116,174)(117,179)(118,178)(119,177)(120,182)(121,181)(122,180)(123,155)
(124,154)(125,153)(126,158)(127,157)(128,156)(129,161)(130,160)(131,159)
(132,164)(133,163)(134,162)(135,167)(136,166)(137,165);
s2 := Sym(182)!(  3,108)(  4,110)(  5,109)(  6,120)(  7,122)(  8,121)(  9,117)
( 10,119)( 11,118)( 12,114)( 13,116)( 14,115)( 15,111)( 16,113)( 17,112)
( 18, 93)( 19, 95)( 20, 94)( 21,105)( 22,107)( 23,106)( 24,102)( 25,104)
( 26,103)( 27, 99)( 28,101)( 29,100)( 30, 96)( 31, 98)( 32, 97)( 33,125)
( 34,124)( 35,123)( 36,137)( 37,136)( 38,135)( 39,134)( 40,133)( 41,132)
( 42,131)( 43,130)( 44,129)( 45,128)( 46,127)( 47,126)( 48,153)( 49,155)
( 50,154)( 51,165)( 52,167)( 53,166)( 54,162)( 55,164)( 56,163)( 57,159)
( 58,161)( 59,160)( 60,156)( 61,158)( 62,157)( 63,138)( 64,140)( 65,139)
( 66,150)( 67,152)( 68,151)( 69,147)( 70,149)( 71,148)( 72,144)( 73,146)
( 74,145)( 75,141)( 76,143)( 77,142)( 78,170)( 79,169)( 80,168)( 81,182)
( 82,181)( 83,180)( 84,179)( 85,178)( 86,177)( 87,176)( 88,175)( 89,174)
( 90,173)( 91,172)( 92,171);
s3 := Sym(182)!(  3,  6)(  4,  7)(  5,  8)(  9, 15)( 10, 16)( 11, 17)( 18, 21)
( 19, 22)( 20, 23)( 24, 30)( 25, 31)( 26, 32)( 33, 36)( 34, 37)( 35, 38)
( 39, 45)( 40, 46)( 41, 47)( 48, 51)( 49, 52)( 50, 53)( 54, 60)( 55, 61)
( 56, 62)( 63, 66)( 64, 67)( 65, 68)( 69, 75)( 70, 76)( 71, 77)( 78, 81)
( 79, 82)( 80, 83)( 84, 90)( 85, 91)( 86, 92)( 93, 96)( 94, 97)( 95, 98)
( 99,105)(100,106)(101,107)(108,111)(109,112)(110,113)(114,120)(115,121)
(116,122)(123,126)(124,127)(125,128)(129,135)(130,136)(131,137)(138,141)
(139,142)(140,143)(144,150)(145,151)(146,152)(153,156)(154,157)(155,158)
(159,165)(160,166)(161,167)(168,171)(169,172)(170,173)(174,180)(175,181)
(176,182);
poly := sub<Sym(182)|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*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*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