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Polytope of Type {14,10,6}

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