Polytope of Type {10,6,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,6,14}*1680
Also Known As : {{10,6|2},{6,14|2}}. if this polytope has another name.
Group : SmallGroup(1680,966)
Rank : 4
Schlafli Type : {10,6,14}
Number of vertices, edges, etc : 10, 30, 42, 14
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 : {10,2,14}*560
   5-fold quotients : {2,6,14}*336
   6-fold quotients : {5,2,14}*280, {10,2,7}*280
   7-fold quotients : {10,6,2}*240
   12-fold quotients : {5,2,7}*140
   15-fold quotients : {2,2,14}*112
   21-fold quotients : {10,2,2}*80
   30-fold quotients : {2,2,7}*56
   35-fold quotients : {2,6,2}*48
   42-fold quotients : {5,2,2}*40
   70-fold quotients : {2,3,2}*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,  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);;
s2 := (  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);;
s3 := (  1,107)(  2,106)(  3,112)(  4,111)(  5,110)(  6,109)(  7,108)(  8,114)
(  9,113)( 10,119)( 11,118)( 12,117)( 13,116)( 14,115)( 15,121)( 16,120)
( 17,126)( 18,125)( 19,124)( 20,123)( 21,122)( 22,128)( 23,127)( 24,133)
( 25,132)( 26,131)( 27,130)( 28,129)( 29,135)( 30,134)( 31,140)( 32,139)
( 33,138)( 34,137)( 35,136)( 36,142)( 37,141)( 38,147)( 39,146)( 40,145)
( 41,144)( 42,143)( 43,149)( 44,148)( 45,154)( 46,153)( 47,152)( 48,151)
( 49,150)( 50,156)( 51,155)( 52,161)( 53,160)( 54,159)( 55,158)( 56,157)
( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)( 64,170)
( 65,169)( 66,175)( 67,174)( 68,173)( 69,172)( 70,171)( 71,177)( 72,176)
( 73,182)( 74,181)( 75,180)( 76,179)( 77,178)( 78,184)( 79,183)( 80,189)
( 81,188)( 82,187)( 83,186)( 84,185)( 85,191)( 86,190)( 87,196)( 88,195)
( 89,194)( 90,193)( 91,192)( 92,198)( 93,197)( 94,203)( 95,202)( 96,201)
( 97,200)( 98,199)( 99,205)(100,204)(101,210)(102,209)(103,208)(104,207)
(105,206);;
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, 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, 
s2*s3*s2*s3*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(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,  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);
s2 := 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);
s3 := Sym(210)!(  1,107)(  2,106)(  3,112)(  4,111)(  5,110)(  6,109)(  7,108)
(  8,114)(  9,113)( 10,119)( 11,118)( 12,117)( 13,116)( 14,115)( 15,121)
( 16,120)( 17,126)( 18,125)( 19,124)( 20,123)( 21,122)( 22,128)( 23,127)
( 24,133)( 25,132)( 26,131)( 27,130)( 28,129)( 29,135)( 30,134)( 31,140)
( 32,139)( 33,138)( 34,137)( 35,136)( 36,142)( 37,141)( 38,147)( 39,146)
( 40,145)( 41,144)( 42,143)( 43,149)( 44,148)( 45,154)( 46,153)( 47,152)
( 48,151)( 49,150)( 50,156)( 51,155)( 52,161)( 53,160)( 54,159)( 55,158)
( 56,157)( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)
( 64,170)( 65,169)( 66,175)( 67,174)( 68,173)( 69,172)( 70,171)( 71,177)
( 72,176)( 73,182)( 74,181)( 75,180)( 76,179)( 77,178)( 78,184)( 79,183)
( 80,189)( 81,188)( 82,187)( 83,186)( 84,185)( 85,191)( 86,190)( 87,196)
( 88,195)( 89,194)( 90,193)( 91,192)( 92,198)( 93,197)( 94,203)( 95,202)
( 96,201)( 97,200)( 98,199)( 99,205)(100,204)(101,210)(102,209)(103,208)
(104,207)(105,206);
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, 
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, 
s2*s3*s2*s3*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.
to this polytope