Polytope of Type {20,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,6,6}*1440b
if this polytope has a name.
Group : SmallGroup(1440,5284)
Rank : 4
Schlafli Type : {20,6,6}
Number of vertices, edges, etc : 20, 60, 18, 6
Order of s₀s₁s₂s₃ : 60
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 : {10,6,6}*720c
   3-fold quotients : {20,6,2}*480a
   5-fold quotients : {4,6,6}*288b
   6-fold quotients : {10,6,2}*240
   9-fold quotients : {20,2,2}*160
   10-fold quotients : {2,6,6}*144c
   15-fold quotients : {4,6,2}*96a
   18-fold quotients : {10,2,2}*80
   20-fold quotients : {2,3,6}*72
   30-fold quotients : {2,6,2}*48
   36-fold quotients : {5,2,2}*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 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)
( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)( 78, 79)
( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 91,136)( 92,140)( 93,139)( 94,138)
( 95,137)( 96,141)( 97,145)( 98,144)( 99,143)(100,142)(101,146)(102,150)
(103,149)(104,148)(105,147)(106,151)(107,155)(108,154)(109,153)(110,152)
(111,156)(112,160)(113,159)(114,158)(115,157)(116,161)(117,165)(118,164)
(119,163)(120,162)(121,166)(122,170)(123,169)(124,168)(125,167)(126,171)
(127,175)(128,174)(129,173)(130,172)(131,176)(132,180)(133,179)(134,178)
(135,177);;
s1 := (  1, 92)(  2, 91)(  3, 95)(  4, 94)(  5, 93)(  6,102)(  7,101)(  8,105)
(  9,104)( 10,103)( 11, 97)( 12, 96)( 13,100)( 14, 99)( 15, 98)( 16,122)
( 17,121)( 18,125)( 19,124)( 20,123)( 21,132)( 22,131)( 23,135)( 24,134)
( 25,133)( 26,127)( 27,126)( 28,130)( 29,129)( 30,128)( 31,107)( 32,106)
( 33,110)( 34,109)( 35,108)( 36,117)( 37,116)( 38,120)( 39,119)( 40,118)
( 41,112)( 42,111)( 43,115)( 44,114)( 45,113)( 46,137)( 47,136)( 48,140)
( 49,139)( 50,138)( 51,147)( 52,146)( 53,150)( 54,149)( 55,148)( 56,142)
( 57,141)( 58,145)( 59,144)( 60,143)( 61,167)( 62,166)( 63,170)( 64,169)
( 65,168)( 66,177)( 67,176)( 68,180)( 69,179)( 70,178)( 71,172)( 72,171)
( 73,175)( 74,174)( 75,173)( 76,152)( 77,151)( 78,155)( 79,154)( 80,153)
( 81,162)( 82,161)( 83,165)( 84,164)( 85,163)( 86,157)( 87,156)( 88,160)
( 89,159)( 90,158);;
s2 := (  1, 21)(  2, 22)(  3, 23)(  4, 24)(  5, 25)(  6, 16)(  7, 17)(  8, 18)
(  9, 19)( 10, 20)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 31, 36)
( 32, 37)( 33, 38)( 34, 39)( 35, 40)( 46, 66)( 47, 67)( 48, 68)( 49, 69)
( 50, 70)( 51, 61)( 52, 62)( 53, 63)( 54, 64)( 55, 65)( 56, 71)( 57, 72)
( 58, 73)( 59, 74)( 60, 75)( 76, 81)( 77, 82)( 78, 83)( 79, 84)( 80, 85)
( 91,111)( 92,112)( 93,113)( 94,114)( 95,115)( 96,106)( 97,107)( 98,108)
( 99,109)(100,110)(101,116)(102,117)(103,118)(104,119)(105,120)(121,126)
(122,127)(123,128)(124,129)(125,130)(136,156)(137,157)(138,158)(139,159)
(140,160)(141,151)(142,152)(143,153)(144,154)(145,155)(146,161)(147,162)
(148,163)(149,164)(150,165)(166,171)(167,172)(168,173)(169,174)(170,175);;
s3 := (  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)( 23, 28)
( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 56)
( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 66, 71)( 67, 72)( 68, 73)( 69, 74)
( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 96,101)( 97,102)
( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)(115,120)
(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)(143,148)
(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)(171,176)
(172,177)(173,178)(174,179)(175,180);;
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*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope