Polytope of Type {14,30}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,30}*840
Also Known As : {14,30|2}. if this polytope has another name.
Group : SmallGroup(840,171)
Rank : 3
Schlafli Type : {14,30}
Number of vertices, edges, etc : 14, 210, 30
Order of s₀s₁s₂ : 210
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,30,2} of size 1680
Vertex Figure Of :
   {2,14,30} of size 1680
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {14,10}*280
   5-fold quotients : {14,6}*168
   7-fold quotients : {2,30}*120
   14-fold quotients : {2,15}*60
   15-fold quotients : {14,2}*56
   21-fold quotients : {2,10}*40
   30-fold quotients : {7,2}*28
   35-fold quotients : {2,6}*24
   42-fold quotients : {2,5}*20
   70-fold quotients : {2,3}*12
   105-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {14,60}*1680, {28,30}*1680a
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, 72)( 37, 71)( 38, 77)( 39, 76)( 40, 75)( 41, 74)( 42, 73)
( 43,100)( 44, 99)( 45,105)( 46,104)( 47,103)( 48,102)( 49,101)( 50, 93)
( 51, 92)( 52, 98)( 53, 97)( 54, 96)( 55, 95)( 56, 94)( 57, 86)( 58, 85)
( 59, 91)( 60, 90)( 61, 89)( 62, 88)( 63, 87)( 64, 79)( 65, 78)( 66, 84)
( 67, 83)( 68, 82)( 69, 81)( 70, 80)(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)(141,177)(142,176)(143,182)
(144,181)(145,180)(146,179)(147,178)(148,205)(149,204)(150,210)(151,209)
(152,208)(153,207)(154,206)(155,198)(156,197)(157,203)(158,202)(159,201)
(160,200)(161,199)(162,191)(163,190)(164,196)(165,195)(166,194)(167,193)
(168,192)(169,184)(170,183)(171,189)(172,188)(173,187)(174,186)(175,185);;
s2 := (  1,148)(  2,149)(  3,150)(  4,151)(  5,152)(  6,153)(  7,154)(  8,141)
(  9,142)( 10,143)( 11,144)( 12,145)( 13,146)( 14,147)( 15,169)( 16,170)
( 17,171)( 18,172)( 19,173)( 20,174)( 21,175)( 22,162)( 23,163)( 24,164)
( 25,165)( 26,166)( 27,167)( 28,168)( 29,155)( 30,156)( 31,157)( 32,158)
( 33,159)( 34,160)( 35,161)( 36,113)( 37,114)( 38,115)( 39,116)( 40,117)
( 41,118)( 42,119)( 43,106)( 44,107)( 45,108)( 46,109)( 47,110)( 48,111)
( 49,112)( 50,134)( 51,135)( 52,136)( 53,137)( 54,138)( 55,139)( 56,140)
( 57,127)( 58,128)( 59,129)( 60,130)( 61,131)( 62,132)( 63,133)( 64,120)
( 65,121)( 66,122)( 67,123)( 68,124)( 69,125)( 70,126)( 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]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*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*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*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*s1*s2*s1*s2 ];;
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, 72)( 37, 71)( 38, 77)( 39, 76)( 40, 75)( 41, 74)
( 42, 73)( 43,100)( 44, 99)( 45,105)( 46,104)( 47,103)( 48,102)( 49,101)
( 50, 93)( 51, 92)( 52, 98)( 53, 97)( 54, 96)( 55, 95)( 56, 94)( 57, 86)
( 58, 85)( 59, 91)( 60, 90)( 61, 89)( 62, 88)( 63, 87)( 64, 79)( 65, 78)
( 66, 84)( 67, 83)( 68, 82)( 69, 81)( 70, 80)(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)(141,177)(142,176)
(143,182)(144,181)(145,180)(146,179)(147,178)(148,205)(149,204)(150,210)
(151,209)(152,208)(153,207)(154,206)(155,198)(156,197)(157,203)(158,202)
(159,201)(160,200)(161,199)(162,191)(163,190)(164,196)(165,195)(166,194)
(167,193)(168,192)(169,184)(170,183)(171,189)(172,188)(173,187)(174,186)
(175,185);
s2 := Sym(210)!(  1,148)(  2,149)(  3,150)(  4,151)(  5,152)(  6,153)(  7,154)
(  8,141)(  9,142)( 10,143)( 11,144)( 12,145)( 13,146)( 14,147)( 15,169)
( 16,170)( 17,171)( 18,172)( 19,173)( 20,174)( 21,175)( 22,162)( 23,163)
( 24,164)( 25,165)( 26,166)( 27,167)( 28,168)( 29,155)( 30,156)( 31,157)
( 32,158)( 33,159)( 34,160)( 35,161)( 36,113)( 37,114)( 38,115)( 39,116)
( 40,117)( 41,118)( 42,119)( 43,106)( 44,107)( 45,108)( 46,109)( 47,110)
( 48,111)( 49,112)( 50,134)( 51,135)( 52,136)( 53,137)( 54,138)( 55,139)
( 56,140)( 57,127)( 58,128)( 59,129)( 60,130)( 61,131)( 62,132)( 63,133)
( 64,120)( 65,121)( 66,122)( 67,123)( 68,124)( 69,125)( 70,126)( 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>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*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*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*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*s1*s2*s1*s2 >;

References : None.
to this polytope