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Polytope of Type {10,14,6}

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