Polytope of Type {2,10,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,10,36}*1440
if this polytope has a name.
Group : SmallGroup(1440,1583)
Rank : 4
Schlafli Type : {2,10,36}
Number of vertices, edges, etc : 2, 10, 180, 36
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 : {2,10,18}*720
   3-fold quotients : {2,10,12}*480
   5-fold quotients : {2,2,36}*288
   6-fold quotients : {2,10,6}*240
   9-fold quotients : {2,10,4}*160
   10-fold quotients : {2,2,18}*144
   15-fold quotients : {2,2,12}*96
   18-fold quotients : {2,10,2}*80
   20-fold quotients : {2,2,9}*72
   30-fold quotients : {2,2,6}*48
   36-fold quotients : {2,5,2}*40
   45-fold quotients : {2,2,4}*32
   60-fold quotients : {2,2,3}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  6, 15)(  7, 16)(  8, 17)(  9, 12)( 10, 13)( 11, 14)( 21, 30)( 22, 31)
( 23, 32)( 24, 27)( 25, 28)( 26, 29)( 36, 45)( 37, 46)( 38, 47)( 39, 42)
( 40, 43)( 41, 44)( 51, 60)( 52, 61)( 53, 62)( 54, 57)( 55, 58)( 56, 59)
( 66, 75)( 67, 76)( 68, 77)( 69, 72)( 70, 73)( 71, 74)( 81, 90)( 82, 91)
( 83, 92)( 84, 87)( 85, 88)( 86, 89)( 96,105)( 97,106)( 98,107)( 99,102)
(100,103)(101,104)(111,120)(112,121)(113,122)(114,117)(115,118)(116,119)
(126,135)(127,136)(128,137)(129,132)(130,133)(131,134)(141,150)(142,151)
(143,152)(144,147)(145,148)(146,149)(156,165)(157,166)(158,167)(159,162)
(160,163)(161,164)(171,180)(172,181)(173,182)(174,177)(175,178)(176,179);;
s2 := (  3,  6)(  4,  8)(  5,  7)(  9, 15)( 10, 17)( 11, 16)( 13, 14)( 18, 38)
( 19, 37)( 20, 36)( 21, 35)( 22, 34)( 23, 33)( 24, 47)( 25, 46)( 26, 45)
( 27, 44)( 28, 43)( 29, 42)( 30, 41)( 31, 40)( 32, 39)( 48, 51)( 49, 53)
( 50, 52)( 54, 60)( 55, 62)( 56, 61)( 58, 59)( 63, 83)( 64, 82)( 65, 81)
( 66, 80)( 67, 79)( 68, 78)( 69, 92)( 70, 91)( 71, 90)( 72, 89)( 73, 88)
( 74, 87)( 75, 86)( 76, 85)( 77, 84)( 93,141)( 94,143)( 95,142)( 96,138)
( 97,140)( 98,139)( 99,150)(100,152)(101,151)(102,147)(103,149)(104,148)
(105,144)(106,146)(107,145)(108,173)(109,172)(110,171)(111,170)(112,169)
(113,168)(114,182)(115,181)(116,180)(117,179)(118,178)(119,177)(120,176)
(121,175)(122,174)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)
(129,167)(130,166)(131,165)(132,164)(133,163)(134,162)(135,161)(136,160)
(137,159);;
s3 := (  3,108)(  4,110)(  5,109)(  6,111)(  7,113)(  8,112)(  9,114)( 10,116)
( 11,115)( 12,117)( 13,119)( 14,118)( 15,120)( 16,122)( 17,121)( 18, 93)
( 19, 95)( 20, 94)( 21, 96)( 22, 98)( 23, 97)( 24, 99)( 25,101)( 26,100)
( 27,102)( 28,104)( 29,103)( 30,105)( 31,107)( 32,106)( 33,125)( 34,124)
( 35,123)( 36,128)( 37,127)( 38,126)( 39,131)( 40,130)( 41,129)( 42,134)
( 43,133)( 44,132)( 45,137)( 46,136)( 47,135)( 48,153)( 49,155)( 50,154)
( 51,156)( 52,158)( 53,157)( 54,159)( 55,161)( 56,160)( 57,162)( 58,164)
( 59,163)( 60,165)( 61,167)( 62,166)( 63,138)( 64,140)( 65,139)( 66,141)
( 67,143)( 68,142)( 69,144)( 70,146)( 71,145)( 72,147)( 73,149)( 74,148)
( 75,150)( 76,152)( 77,151)( 78,170)( 79,169)( 80,168)( 81,173)( 82,172)
( 83,171)( 84,176)( 85,175)( 86,174)( 87,179)( 88,178)( 89,177)( 90,182)
( 91,181)( 92,180);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(182)!(1,2);
s1 := Sym(182)!(  6, 15)(  7, 16)(  8, 17)(  9, 12)( 10, 13)( 11, 14)( 21, 30)
( 22, 31)( 23, 32)( 24, 27)( 25, 28)( 26, 29)( 36, 45)( 37, 46)( 38, 47)
( 39, 42)( 40, 43)( 41, 44)( 51, 60)( 52, 61)( 53, 62)( 54, 57)( 55, 58)
( 56, 59)( 66, 75)( 67, 76)( 68, 77)( 69, 72)( 70, 73)( 71, 74)( 81, 90)
( 82, 91)( 83, 92)( 84, 87)( 85, 88)( 86, 89)( 96,105)( 97,106)( 98,107)
( 99,102)(100,103)(101,104)(111,120)(112,121)(113,122)(114,117)(115,118)
(116,119)(126,135)(127,136)(128,137)(129,132)(130,133)(131,134)(141,150)
(142,151)(143,152)(144,147)(145,148)(146,149)(156,165)(157,166)(158,167)
(159,162)(160,163)(161,164)(171,180)(172,181)(173,182)(174,177)(175,178)
(176,179);
s2 := Sym(182)!(  3,  6)(  4,  8)(  5,  7)(  9, 15)( 10, 17)( 11, 16)( 13, 14)
( 18, 38)( 19, 37)( 20, 36)( 21, 35)( 22, 34)( 23, 33)( 24, 47)( 25, 46)
( 26, 45)( 27, 44)( 28, 43)( 29, 42)( 30, 41)( 31, 40)( 32, 39)( 48, 51)
( 49, 53)( 50, 52)( 54, 60)( 55, 62)( 56, 61)( 58, 59)( 63, 83)( 64, 82)
( 65, 81)( 66, 80)( 67, 79)( 68, 78)( 69, 92)( 70, 91)( 71, 90)( 72, 89)
( 73, 88)( 74, 87)( 75, 86)( 76, 85)( 77, 84)( 93,141)( 94,143)( 95,142)
( 96,138)( 97,140)( 98,139)( 99,150)(100,152)(101,151)(102,147)(103,149)
(104,148)(105,144)(106,146)(107,145)(108,173)(109,172)(110,171)(111,170)
(112,169)(113,168)(114,182)(115,181)(116,180)(117,179)(118,178)(119,177)
(120,176)(121,175)(122,174)(123,158)(124,157)(125,156)(126,155)(127,154)
(128,153)(129,167)(130,166)(131,165)(132,164)(133,163)(134,162)(135,161)
(136,160)(137,159);
s3 := Sym(182)!(  3,108)(  4,110)(  5,109)(  6,111)(  7,113)(  8,112)(  9,114)
( 10,116)( 11,115)( 12,117)( 13,119)( 14,118)( 15,120)( 16,122)( 17,121)
( 18, 93)( 19, 95)( 20, 94)( 21, 96)( 22, 98)( 23, 97)( 24, 99)( 25,101)
( 26,100)( 27,102)( 28,104)( 29,103)( 30,105)( 31,107)( 32,106)( 33,125)
( 34,124)( 35,123)( 36,128)( 37,127)( 38,126)( 39,131)( 40,130)( 41,129)
( 42,134)( 43,133)( 44,132)( 45,137)( 46,136)( 47,135)( 48,153)( 49,155)
( 50,154)( 51,156)( 52,158)( 53,157)( 54,159)( 55,161)( 56,160)( 57,162)
( 58,164)( 59,163)( 60,165)( 61,167)( 62,166)( 63,138)( 64,140)( 65,139)
( 66,141)( 67,143)( 68,142)( 69,144)( 70,146)( 71,145)( 72,147)( 73,149)
( 74,148)( 75,150)( 76,152)( 77,151)( 78,170)( 79,169)( 80,168)( 81,173)
( 82,172)( 83,171)( 84,176)( 85,175)( 86,174)( 87,179)( 88,178)( 89,177)
( 90,182)( 91,181)( 92,180);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*s2*s3*s2*s3 >;

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