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Polytope of Type {6,14,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,14,10}*1680
Also Known As : {{6,14|2},{14,10|2}}. if this polytope has another name.
Group : SmallGroup(1680,966)
Rank : 4
Schlafli Type : {6,14,10}
Number of vertices, edges, etc : 6, 42, 70, 10
Order of s0s1s2s3 : 210
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) :
   3-fold quotients : {2,14,10}*560
   5-fold quotients : {6,14,2}*336
   7-fold quotients : {6,2,10}*240
   14-fold quotients : {3,2,10}*120, {6,2,5}*120
   15-fold quotients : {2,14,2}*112
   21-fold quotients : {2,2,10}*80
   28-fold quotients : {3,2,5}*60
   30-fold quotients : {2,7,2}*56
   35-fold quotients : {6,2,2}*48
   42-fold quotients : {2,2,5}*40
   70-fold quotients : {3,2,2}*24
   105-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 36, 71)( 37, 72)( 38, 73)( 39, 74)( 40, 75)( 41, 76)( 42, 77)( 43, 78)
( 44, 79)( 45, 80)( 46, 81)( 47, 82)( 48, 83)( 49, 84)( 50, 85)( 51, 86)
( 52, 87)( 53, 88)( 54, 89)( 55, 90)( 56, 91)( 57, 92)( 58, 93)( 59, 94)
( 60, 95)( 61, 96)( 62, 97)( 63, 98)( 64, 99)( 65,100)( 66,101)( 67,102)
( 68,103)( 69,104)( 70,105)(141,176)(142,177)(143,178)(144,179)(145,180)
(146,181)(147,182)(148,183)(149,184)(150,185)(151,186)(152,187)(153,188)
(154,189)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)(161,196)
(162,197)(163,198)(164,199)(165,200)(166,201)(167,202)(168,203)(169,204)
(170,205)(171,206)(172,207)(173,208)(174,209)(175,210);;
s1 := (  1, 36)(  2, 42)(  3, 41)(  4, 40)(  5, 39)(  6, 38)(  7, 37)(  8, 43)
(  9, 49)( 10, 48)( 11, 47)( 12, 46)( 13, 45)( 14, 44)( 15, 50)( 16, 56)
( 17, 55)( 18, 54)( 19, 53)( 20, 52)( 21, 51)( 22, 57)( 23, 63)( 24, 62)
( 25, 61)( 26, 60)( 27, 59)( 28, 58)( 29, 64)( 30, 70)( 31, 69)( 32, 68)
( 33, 67)( 34, 66)( 35, 65)( 72, 77)( 73, 76)( 74, 75)( 79, 84)( 80, 83)
( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)( 94, 97)( 95, 96)(100,105)
(101,104)(102,103)(106,141)(107,147)(108,146)(109,145)(110,144)(111,143)
(112,142)(113,148)(114,154)(115,153)(116,152)(117,151)(118,150)(119,149)
(120,155)(121,161)(122,160)(123,159)(124,158)(125,157)(126,156)(127,162)
(128,168)(129,167)(130,166)(131,165)(132,164)(133,163)(134,169)(135,175)
(136,174)(137,173)(138,172)(139,171)(140,170)(177,182)(178,181)(179,180)
(184,189)(185,188)(186,187)(191,196)(192,195)(193,194)(198,203)(199,202)
(200,201)(205,210)(206,209)(207,208);;
s2 := (  1,  2)(  3,  7)(  4,  6)(  8, 30)(  9, 29)( 10, 35)( 11, 34)( 12, 33)
( 13, 32)( 14, 31)( 15, 23)( 16, 22)( 17, 28)( 18, 27)( 19, 26)( 20, 25)
( 21, 24)( 36, 37)( 38, 42)( 39, 41)( 43, 65)( 44, 64)( 45, 70)( 46, 69)
( 47, 68)( 48, 67)( 49, 66)( 50, 58)( 51, 57)( 52, 63)( 53, 62)( 54, 61)
( 55, 60)( 56, 59)( 71, 72)( 73, 77)( 74, 76)( 78,100)( 79, 99)( 80,105)
( 81,104)( 82,103)( 83,102)( 84,101)( 85, 93)( 86, 92)( 87, 98)( 88, 97)
( 89, 96)( 90, 95)( 91, 94)(106,107)(108,112)(109,111)(113,135)(114,134)
(115,140)(116,139)(117,138)(118,137)(119,136)(120,128)(121,127)(122,133)
(123,132)(124,131)(125,130)(126,129)(141,142)(143,147)(144,146)(148,170)
(149,169)(150,175)(151,174)(152,173)(153,172)(154,171)(155,163)(156,162)
(157,168)(158,167)(159,166)(160,165)(161,164)(176,177)(178,182)(179,181)
(183,205)(184,204)(185,210)(186,209)(187,208)(188,207)(189,206)(190,198)
(191,197)(192,203)(193,202)(194,201)(195,200)(196,199);;
s3 := (  1,113)(  2,114)(  3,115)(  4,116)(  5,117)(  6,118)(  7,119)(  8,106)
(  9,107)( 10,108)( 11,109)( 12,110)( 13,111)( 14,112)( 15,134)( 16,135)
( 17,136)( 18,137)( 19,138)( 20,139)( 21,140)( 22,127)( 23,128)( 24,129)
( 25,130)( 26,131)( 27,132)( 28,133)( 29,120)( 30,121)( 31,122)( 32,123)
( 33,124)( 34,125)( 35,126)( 36,148)( 37,149)( 38,150)( 39,151)( 40,152)
( 41,153)( 42,154)( 43,141)( 44,142)( 45,143)( 46,144)( 47,145)( 48,146)
( 49,147)( 50,169)( 51,170)( 52,171)( 53,172)( 54,173)( 55,174)( 56,175)
( 57,162)( 58,163)( 59,164)( 60,165)( 61,166)( 62,167)( 63,168)( 64,155)
( 65,156)( 66,157)( 67,158)( 68,159)( 69,160)( 70,161)( 71,183)( 72,184)
( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,176)( 79,177)( 80,178)
( 81,179)( 82,180)( 83,181)( 84,182)( 85,204)( 86,205)( 87,206)( 88,207)
( 89,208)( 90,209)( 91,210)( 92,197)( 93,198)( 94,199)( 95,200)( 96,201)
( 97,202)( 98,203)( 99,190)(100,191)(101,192)(102,193)(103,194)(104,195)
(105,196);;
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, 
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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(210)!( 36, 71)( 37, 72)( 38, 73)( 39, 74)( 40, 75)( 41, 76)( 42, 77)
( 43, 78)( 44, 79)( 45, 80)( 46, 81)( 47, 82)( 48, 83)( 49, 84)( 50, 85)
( 51, 86)( 52, 87)( 53, 88)( 54, 89)( 55, 90)( 56, 91)( 57, 92)( 58, 93)
( 59, 94)( 60, 95)( 61, 96)( 62, 97)( 63, 98)( 64, 99)( 65,100)( 66,101)
( 67,102)( 68,103)( 69,104)( 70,105)(141,176)(142,177)(143,178)(144,179)
(145,180)(146,181)(147,182)(148,183)(149,184)(150,185)(151,186)(152,187)
(153,188)(154,189)(155,190)(156,191)(157,192)(158,193)(159,194)(160,195)
(161,196)(162,197)(163,198)(164,199)(165,200)(166,201)(167,202)(168,203)
(169,204)(170,205)(171,206)(172,207)(173,208)(174,209)(175,210);
s1 := Sym(210)!(  1, 36)(  2, 42)(  3, 41)(  4, 40)(  5, 39)(  6, 38)(  7, 37)
(  8, 43)(  9, 49)( 10, 48)( 11, 47)( 12, 46)( 13, 45)( 14, 44)( 15, 50)
( 16, 56)( 17, 55)( 18, 54)( 19, 53)( 20, 52)( 21, 51)( 22, 57)( 23, 63)
( 24, 62)( 25, 61)( 26, 60)( 27, 59)( 28, 58)( 29, 64)( 30, 70)( 31, 69)
( 32, 68)( 33, 67)( 34, 66)( 35, 65)( 72, 77)( 73, 76)( 74, 75)( 79, 84)
( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)( 94, 97)( 95, 96)
(100,105)(101,104)(102,103)(106,141)(107,147)(108,146)(109,145)(110,144)
(111,143)(112,142)(113,148)(114,154)(115,153)(116,152)(117,151)(118,150)
(119,149)(120,155)(121,161)(122,160)(123,159)(124,158)(125,157)(126,156)
(127,162)(128,168)(129,167)(130,166)(131,165)(132,164)(133,163)(134,169)
(135,175)(136,174)(137,173)(138,172)(139,171)(140,170)(177,182)(178,181)
(179,180)(184,189)(185,188)(186,187)(191,196)(192,195)(193,194)(198,203)
(199,202)(200,201)(205,210)(206,209)(207,208);
s2 := Sym(210)!(  1,  2)(  3,  7)(  4,  6)(  8, 30)(  9, 29)( 10, 35)( 11, 34)
( 12, 33)( 13, 32)( 14, 31)( 15, 23)( 16, 22)( 17, 28)( 18, 27)( 19, 26)
( 20, 25)( 21, 24)( 36, 37)( 38, 42)( 39, 41)( 43, 65)( 44, 64)( 45, 70)
( 46, 69)( 47, 68)( 48, 67)( 49, 66)( 50, 58)( 51, 57)( 52, 63)( 53, 62)
( 54, 61)( 55, 60)( 56, 59)( 71, 72)( 73, 77)( 74, 76)( 78,100)( 79, 99)
( 80,105)( 81,104)( 82,103)( 83,102)( 84,101)( 85, 93)( 86, 92)( 87, 98)
( 88, 97)( 89, 96)( 90, 95)( 91, 94)(106,107)(108,112)(109,111)(113,135)
(114,134)(115,140)(116,139)(117,138)(118,137)(119,136)(120,128)(121,127)
(122,133)(123,132)(124,131)(125,130)(126,129)(141,142)(143,147)(144,146)
(148,170)(149,169)(150,175)(151,174)(152,173)(153,172)(154,171)(155,163)
(156,162)(157,168)(158,167)(159,166)(160,165)(161,164)(176,177)(178,182)
(179,181)(183,205)(184,204)(185,210)(186,209)(187,208)(188,207)(189,206)
(190,198)(191,197)(192,203)(193,202)(194,201)(195,200)(196,199);
s3 := Sym(210)!(  1,113)(  2,114)(  3,115)(  4,116)(  5,117)(  6,118)(  7,119)
(  8,106)(  9,107)( 10,108)( 11,109)( 12,110)( 13,111)( 14,112)( 15,134)
( 16,135)( 17,136)( 18,137)( 19,138)( 20,139)( 21,140)( 22,127)( 23,128)
( 24,129)( 25,130)( 26,131)( 27,132)( 28,133)( 29,120)( 30,121)( 31,122)
( 32,123)( 33,124)( 34,125)( 35,126)( 36,148)( 37,149)( 38,150)( 39,151)
( 40,152)( 41,153)( 42,154)( 43,141)( 44,142)( 45,143)( 46,144)( 47,145)
( 48,146)( 49,147)( 50,169)( 51,170)( 52,171)( 53,172)( 54,173)( 55,174)
( 56,175)( 57,162)( 58,163)( 59,164)( 60,165)( 61,166)( 62,167)( 63,168)
( 64,155)( 65,156)( 66,157)( 67,158)( 68,159)( 69,160)( 70,161)( 71,183)
( 72,184)( 73,185)( 74,186)( 75,187)( 76,188)( 77,189)( 78,176)( 79,177)
( 80,178)( 81,179)( 82,180)( 83,181)( 84,182)( 85,204)( 86,205)( 87,206)
( 88,207)( 89,208)( 90,209)( 91,210)( 92,197)( 93,198)( 94,199)( 95,200)
( 96,201)( 97,202)( 98,203)( 99,190)(100,191)(101,192)(102,193)(103,194)
(104,195)(105,196);
poly := sub<Sym(210)|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, 
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 >; 
 
References : None.
to this polytope