Polytope of Type {6,80}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,80}*960
Also Known As : {6,80|2}. if this polytope has another name.
Group : SmallGroup(960,1013)
Rank : 3
Schlafli Type : {6,80}
Number of vertices, edges, etc : 6, 240, 80
Order of s₀s₁s₂ : 240
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,80,2} of size 1920
Vertex Figure Of :
   {2,6,80} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,40}*480
   3-fold quotients : {2,80}*320
   4-fold quotients : {6,20}*240a
   5-fold quotients : {6,16}*192
   6-fold quotients : {2,40}*160
   8-fold quotients : {6,10}*120
   10-fold quotients : {6,8}*96
   12-fold quotients : {2,20}*80
   15-fold quotients : {2,16}*64
   20-fold quotients : {6,4}*48a
   24-fold quotients : {2,10}*40
   30-fold quotients : {2,8}*32
   40-fold quotients : {6,2}*24
   48-fold quotients : {2,5}*20
   60-fold quotients : {2,4}*16
   80-fold quotients : {3,2}*12
   120-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,80}*1920a, {6,160}*1920
Permutation Representation (GAP) :

s0 := (  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)( 23, 28)
( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 51, 56)
( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 66, 71)( 67, 72)( 68, 73)( 69, 74)
( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 96,101)( 97,102)
( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)(115,120)
(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)(143,148)
(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)(171,176)
(172,177)(173,178)(174,179)(175,180)(186,191)(187,192)(188,193)(189,194)
(190,195)(201,206)(202,207)(203,208)(204,209)(205,210)(216,221)(217,222)
(218,223)(219,224)(220,225)(231,236)(232,237)(233,238)(234,239)(235,240);;
s1 := (  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 12, 15)( 13, 14)( 16, 21)
( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 51)( 32, 55)
( 33, 54)( 34, 53)( 35, 52)( 36, 46)( 37, 50)( 38, 49)( 39, 48)( 40, 47)
( 41, 56)( 42, 60)( 43, 59)( 44, 58)( 45, 57)( 61, 96)( 62,100)( 63, 99)
( 64, 98)( 65, 97)( 66, 91)( 67, 95)( 68, 94)( 69, 93)( 70, 92)( 71,101)
( 72,105)( 73,104)( 74,103)( 75,102)( 76,111)( 77,115)( 78,114)( 79,113)
( 80,112)( 81,106)( 82,110)( 83,109)( 84,108)( 85,107)( 86,116)( 87,120)
( 88,119)( 89,118)( 90,117)(121,186)(122,190)(123,189)(124,188)(125,187)
(126,181)(127,185)(128,184)(129,183)(130,182)(131,191)(132,195)(133,194)
(134,193)(135,192)(136,201)(137,205)(138,204)(139,203)(140,202)(141,196)
(142,200)(143,199)(144,198)(145,197)(146,206)(147,210)(148,209)(149,208)
(150,207)(151,231)(152,235)(153,234)(154,233)(155,232)(156,226)(157,230)
(158,229)(159,228)(160,227)(161,236)(162,240)(163,239)(164,238)(165,237)
(166,216)(167,220)(168,219)(169,218)(170,217)(171,211)(172,215)(173,214)
(174,213)(175,212)(176,221)(177,225)(178,224)(179,223)(180,222);;
s2 := (  1,122)(  2,121)(  3,125)(  4,124)(  5,123)(  6,127)(  7,126)(  8,130)
(  9,129)( 10,128)( 11,132)( 12,131)( 13,135)( 14,134)( 15,133)( 16,137)
( 17,136)( 18,140)( 19,139)( 20,138)( 21,142)( 22,141)( 23,145)( 24,144)
( 25,143)( 26,147)( 27,146)( 28,150)( 29,149)( 30,148)( 31,167)( 32,166)
( 33,170)( 34,169)( 35,168)( 36,172)( 37,171)( 38,175)( 39,174)( 40,173)
( 41,177)( 42,176)( 43,180)( 44,179)( 45,178)( 46,152)( 47,151)( 48,155)
( 49,154)( 50,153)( 51,157)( 52,156)( 53,160)( 54,159)( 55,158)( 56,162)
( 57,161)( 58,165)( 59,164)( 60,163)( 61,212)( 62,211)( 63,215)( 64,214)
( 65,213)( 66,217)( 67,216)( 68,220)( 69,219)( 70,218)( 71,222)( 72,221)
( 73,225)( 74,224)( 75,223)( 76,227)( 77,226)( 78,230)( 79,229)( 80,228)
( 81,232)( 82,231)( 83,235)( 84,234)( 85,233)( 86,237)( 87,236)( 88,240)
( 89,239)( 90,238)( 91,182)( 92,181)( 93,185)( 94,184)( 95,183)( 96,187)
( 97,186)( 98,190)( 99,189)(100,188)(101,192)(102,191)(103,195)(104,194)
(105,193)(106,197)(107,196)(108,200)(109,199)(110,198)(111,202)(112,201)
(113,205)(114,204)(115,203)(116,207)(117,206)(118,210)(119,209)(120,208);;
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, 
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*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*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(240)!(  6, 11)(  7, 12)(  8, 13)(  9, 14)( 10, 15)( 21, 26)( 22, 27)
( 23, 28)( 24, 29)( 25, 30)( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)
( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 66, 71)( 67, 72)( 68, 73)
( 69, 74)( 70, 75)( 81, 86)( 82, 87)( 83, 88)( 84, 89)( 85, 90)( 96,101)
( 97,102)( 98,103)( 99,104)(100,105)(111,116)(112,117)(113,118)(114,119)
(115,120)(126,131)(127,132)(128,133)(129,134)(130,135)(141,146)(142,147)
(143,148)(144,149)(145,150)(156,161)(157,162)(158,163)(159,164)(160,165)
(171,176)(172,177)(173,178)(174,179)(175,180)(186,191)(187,192)(188,193)
(189,194)(190,195)(201,206)(202,207)(203,208)(204,209)(205,210)(216,221)
(217,222)(218,223)(219,224)(220,225)(231,236)(232,237)(233,238)(234,239)
(235,240);
s1 := Sym(240)!(  1,  6)(  2, 10)(  3,  9)(  4,  8)(  5,  7)( 12, 15)( 13, 14)
( 16, 21)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 30)( 28, 29)( 31, 51)
( 32, 55)( 33, 54)( 34, 53)( 35, 52)( 36, 46)( 37, 50)( 38, 49)( 39, 48)
( 40, 47)( 41, 56)( 42, 60)( 43, 59)( 44, 58)( 45, 57)( 61, 96)( 62,100)
( 63, 99)( 64, 98)( 65, 97)( 66, 91)( 67, 95)( 68, 94)( 69, 93)( 70, 92)
( 71,101)( 72,105)( 73,104)( 74,103)( 75,102)( 76,111)( 77,115)( 78,114)
( 79,113)( 80,112)( 81,106)( 82,110)( 83,109)( 84,108)( 85,107)( 86,116)
( 87,120)( 88,119)( 89,118)( 90,117)(121,186)(122,190)(123,189)(124,188)
(125,187)(126,181)(127,185)(128,184)(129,183)(130,182)(131,191)(132,195)
(133,194)(134,193)(135,192)(136,201)(137,205)(138,204)(139,203)(140,202)
(141,196)(142,200)(143,199)(144,198)(145,197)(146,206)(147,210)(148,209)
(149,208)(150,207)(151,231)(152,235)(153,234)(154,233)(155,232)(156,226)
(157,230)(158,229)(159,228)(160,227)(161,236)(162,240)(163,239)(164,238)
(165,237)(166,216)(167,220)(168,219)(169,218)(170,217)(171,211)(172,215)
(173,214)(174,213)(175,212)(176,221)(177,225)(178,224)(179,223)(180,222);
s2 := Sym(240)!(  1,122)(  2,121)(  3,125)(  4,124)(  5,123)(  6,127)(  7,126)
(  8,130)(  9,129)( 10,128)( 11,132)( 12,131)( 13,135)( 14,134)( 15,133)
( 16,137)( 17,136)( 18,140)( 19,139)( 20,138)( 21,142)( 22,141)( 23,145)
( 24,144)( 25,143)( 26,147)( 27,146)( 28,150)( 29,149)( 30,148)( 31,167)
( 32,166)( 33,170)( 34,169)( 35,168)( 36,172)( 37,171)( 38,175)( 39,174)
( 40,173)( 41,177)( 42,176)( 43,180)( 44,179)( 45,178)( 46,152)( 47,151)
( 48,155)( 49,154)( 50,153)( 51,157)( 52,156)( 53,160)( 54,159)( 55,158)
( 56,162)( 57,161)( 58,165)( 59,164)( 60,163)( 61,212)( 62,211)( 63,215)
( 64,214)( 65,213)( 66,217)( 67,216)( 68,220)( 69,219)( 70,218)( 71,222)
( 72,221)( 73,225)( 74,224)( 75,223)( 76,227)( 77,226)( 78,230)( 79,229)
( 80,228)( 81,232)( 82,231)( 83,235)( 84,234)( 85,233)( 86,237)( 87,236)
( 88,240)( 89,239)( 90,238)( 91,182)( 92,181)( 93,185)( 94,184)( 95,183)
( 96,187)( 97,186)( 98,190)( 99,189)(100,188)(101,192)(102,191)(103,195)
(104,194)(105,193)(106,197)(107,196)(108,200)(109,199)(110,198)(111,202)
(112,201)(113,205)(114,204)(115,203)(116,207)(117,206)(118,210)(119,209)
(120,208);
poly := sub<Sym(240)|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, 
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*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*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