Polytope of Type {6,10,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,10,14}*1680
Also Known As : {{6,10|2},{10,14|2}}. if this polytope has another name.
Group : SmallGroup(1680,966)
Rank : 4
Schlafli Type : {6,10,14}
Number of vertices, edges, etc : 6, 30, 70, 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 : {2,10,14}*560
   5-fold quotients : {6,2,14}*336
   7-fold quotients : {6,10,2}*240
   10-fold quotients : {3,2,14}*168, {6,2,7}*168
   15-fold quotients : {2,2,14}*112
   20-fold quotients : {3,2,7}*84
   21-fold quotients : {2,10,2}*80
   30-fold quotients : {2,2,7}*56
   35-fold quotients : {6,2,2}*48
   42-fold quotients : {2,5,2}*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, 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);;
s2 := (  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);;
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, 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, 
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)!( 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, 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);
s2 := 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);
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, 
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, 
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