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Polytope of Type {6,10,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,10,12}*1440
Also Known As : {{6,10|2},{10,12|2}}. if this polytope has another name.
Group : SmallGroup(1440,5282)
Rank : 4
Schlafli Type : {6,10,12}
Number of vertices, edges, etc : 6, 30, 60, 12
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 : {6,10,6}*720
   3-fold quotients : {2,10,12}*480, {6,10,4}*480
   5-fold quotients : {6,2,12}*288
   6-fold quotients : {2,10,6}*240, {6,10,2}*240
   9-fold quotients : {2,10,4}*160
   10-fold quotients : {3,2,12}*144, {6,2,6}*144
   15-fold quotients : {2,2,12}*96, {6,2,4}*96
   18-fold quotients : {2,10,2}*80
   20-fold quotients : {3,2,6}*72, {6,2,3}*72
   30-fold quotients : {3,2,4}*48, {2,2,6}*48, {6,2,2}*48
   36-fold quotients : {2,5,2}*40
   40-fold quotients : {3,2,3}*36
   45-fold quotients : {2,2,4}*32
   60-fold quotients : {2,2,3}*24, {3,2,2}*24
   90-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  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);;
s1 := (  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 12, 15)( 13, 14)( 16, 21)
( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 36)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 51)( 47, 55)( 48, 54)
( 49, 53)( 50, 52)( 57, 60)( 58, 59)( 61, 66)( 62, 70)( 63, 69)( 64, 68)
( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)( 80, 82)
( 87, 90)( 88, 89)( 91, 96)( 92,100)( 93, 99)( 94, 98)( 95, 97)(102,105)
(103,104)(106,111)(107,115)(108,114)(109,113)(110,112)(117,120)(118,119)
(121,126)(122,130)(123,129)(124,128)(125,127)(132,135)(133,134)(136,141)
(137,145)(138,144)(139,143)(140,142)(147,150)(148,149)(151,156)(152,160)
(153,159)(154,158)(155,157)(162,165)(163,164)(166,171)(167,175)(168,174)
(169,173)(170,172)(177,180)(178,179);;
s2 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 32)( 17, 31)
( 18, 35)( 19, 34)( 20, 33)( 21, 37)( 22, 36)( 23, 40)( 24, 39)( 25, 38)
( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 47)( 48, 50)( 51, 52)
( 53, 55)( 56, 57)( 58, 60)( 61, 77)( 62, 76)( 63, 80)( 64, 79)( 65, 78)
( 66, 82)( 67, 81)( 68, 85)( 69, 84)( 70, 83)( 71, 87)( 72, 86)( 73, 90)
( 74, 89)( 75, 88)( 91,137)( 92,136)( 93,140)( 94,139)( 95,138)( 96,142)
( 97,141)( 98,145)( 99,144)(100,143)(101,147)(102,146)(103,150)(104,149)
(105,148)(106,167)(107,166)(108,170)(109,169)(110,168)(111,172)(112,171)
(113,175)(114,174)(115,173)(116,177)(117,176)(118,180)(119,179)(120,178)
(121,152)(122,151)(123,155)(124,154)(125,153)(126,157)(127,156)(128,160)
(129,159)(130,158)(131,162)(132,161)(133,165)(134,164)(135,163);;
s3 := (  1,106)(  2,107)(  3,108)(  4,109)(  5,110)(  6,111)(  7,112)(  8,113)
(  9,114)( 10,115)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)( 16, 91)
( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 96)( 22, 97)( 23, 98)( 24, 99)
( 25,100)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,121)( 32,122)
( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)( 40,130)
( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,151)( 47,152)( 48,153)
( 49,154)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)( 56,161)
( 57,162)( 58,163)( 59,164)( 60,165)( 61,136)( 62,137)( 63,138)( 64,139)
( 65,140)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,146)( 72,147)
( 73,148)( 74,149)( 75,150)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)
( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)
( 89,179)( 90,180);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(180)!(  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 12, 15)( 13, 14)
( 16, 21)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 36)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 45)( 43, 44)( 46, 51)( 47, 55)
( 48, 54)( 49, 53)( 50, 52)( 57, 60)( 58, 59)( 61, 66)( 62, 70)( 63, 69)
( 64, 68)( 65, 67)( 72, 75)( 73, 74)( 76, 81)( 77, 85)( 78, 84)( 79, 83)
( 80, 82)( 87, 90)( 88, 89)( 91, 96)( 92,100)( 93, 99)( 94, 98)( 95, 97)
(102,105)(103,104)(106,111)(107,115)(108,114)(109,113)(110,112)(117,120)
(118,119)(121,126)(122,130)(123,129)(124,128)(125,127)(132,135)(133,134)
(136,141)(137,145)(138,144)(139,143)(140,142)(147,150)(148,149)(151,156)
(152,160)(153,159)(154,158)(155,157)(162,165)(163,164)(166,171)(167,175)
(168,174)(169,173)(170,172)(177,180)(178,179);
s2 := Sym(180)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 32)
( 17, 31)( 18, 35)( 19, 34)( 20, 33)( 21, 37)( 22, 36)( 23, 40)( 24, 39)
( 25, 38)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 47)( 48, 50)
( 51, 52)( 53, 55)( 56, 57)( 58, 60)( 61, 77)( 62, 76)( 63, 80)( 64, 79)
( 65, 78)( 66, 82)( 67, 81)( 68, 85)( 69, 84)( 70, 83)( 71, 87)( 72, 86)
( 73, 90)( 74, 89)( 75, 88)( 91,137)( 92,136)( 93,140)( 94,139)( 95,138)
( 96,142)( 97,141)( 98,145)( 99,144)(100,143)(101,147)(102,146)(103,150)
(104,149)(105,148)(106,167)(107,166)(108,170)(109,169)(110,168)(111,172)
(112,171)(113,175)(114,174)(115,173)(116,177)(117,176)(118,180)(119,179)
(120,178)(121,152)(122,151)(123,155)(124,154)(125,153)(126,157)(127,156)
(128,160)(129,159)(130,158)(131,162)(132,161)(133,165)(134,164)(135,163);
s3 := Sym(180)!(  1,106)(  2,107)(  3,108)(  4,109)(  5,110)(  6,111)(  7,112)
(  8,113)(  9,114)( 10,115)( 11,116)( 12,117)( 13,118)( 14,119)( 15,120)
( 16, 91)( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 96)( 22, 97)( 23, 98)
( 24, 99)( 25,100)( 26,101)( 27,102)( 28,103)( 29,104)( 30,105)( 31,121)
( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)
( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,151)( 47,152)
( 48,153)( 49,154)( 50,155)( 51,156)( 52,157)( 53,158)( 54,159)( 55,160)
( 56,161)( 57,162)( 58,163)( 59,164)( 60,165)( 61,136)( 62,137)( 63,138)
( 64,139)( 65,140)( 66,141)( 67,142)( 68,143)( 69,144)( 70,145)( 71,146)
( 72,147)( 73,148)( 74,149)( 75,150)( 76,166)( 77,167)( 78,168)( 79,169)
( 80,170)( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)
( 88,178)( 89,179)( 90,180);
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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 >; 
 
References : None.
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