Polytope of Type {14,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,6,10}*1680
Also Known As : {{14,6|2},{6,10|2}}. if this polytope has another name.
Group : SmallGroup(1680,966)
Rank : 4
Schlafli Type : {14,6,10}
Number of vertices, edges, etc : 14, 42, 30, 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 : {14,2,10}*560
   5-fold quotients : {14,6,2}*336
   6-fold quotients : {7,2,10}*280, {14,2,5}*280
   7-fold quotients : {2,6,10}*240
   12-fold quotients : {7,2,5}*140
   15-fold quotients : {14,2,2}*112
   21-fold quotients : {2,2,10}*80
   30-fold quotients : {7,2,2}*56
   35-fold quotients : {2,6,2}*48
   42-fold quotients : {2,2,5}*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 := (  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,  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);;
s2 := (  1, 36)(  2, 37)(  3, 38)(  4, 39)(  5, 40)(  6, 41)(  7, 42)(  8, 64)
(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 57)( 16, 58)
( 17, 59)( 18, 60)( 19, 61)( 20, 62)( 21, 63)( 22, 50)( 23, 51)( 24, 52)
( 25, 53)( 26, 54)( 27, 55)( 28, 56)( 29, 43)( 30, 44)( 31, 45)( 32, 46)
( 33, 47)( 34, 48)( 35, 49)( 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)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)(112,147)
(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)(120,162)
(121,163)(122,164)(123,165)(124,166)(125,167)(126,168)(127,155)(128,156)
(129,157)(130,158)(131,159)(132,160)(133,161)(134,148)(135,149)(136,150)
(137,151)(138,152)(139,153)(140,154)(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);;
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, 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, 
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,  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);
s2 := Sym(210)!(  1, 36)(  2, 37)(  3, 38)(  4, 39)(  5, 40)(  6, 41)(  7, 42)
(  8, 64)(  9, 65)( 10, 66)( 11, 67)( 12, 68)( 13, 69)( 14, 70)( 15, 57)
( 16, 58)( 17, 59)( 18, 60)( 19, 61)( 20, 62)( 21, 63)( 22, 50)( 23, 51)
( 24, 52)( 25, 53)( 26, 54)( 27, 55)( 28, 56)( 29, 43)( 30, 44)( 31, 45)
( 32, 46)( 33, 47)( 34, 48)( 35, 49)( 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)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)
(112,147)(113,169)(114,170)(115,171)(116,172)(117,173)(118,174)(119,175)
(120,162)(121,163)(122,164)(123,165)(124,166)(125,167)(126,168)(127,155)
(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,148)(135,149)
(136,150)(137,151)(138,152)(139,153)(140,154)(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);
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, 
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, 
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