Polytope of Type {80,6}

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

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