Polytope of Type {2,80,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,80,6}*1920
if this polytope has a name.
Group : SmallGroup(1920,203907)
Rank : 4
Schlafli Type : {2,80,6}
Number of vertices, edges, etc : 2, 80, 240, 6
Order of s₀s₁s₂s₃ : 240
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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) :
   2-fold quotients : {2,40,6}*960
   3-fold quotients : {2,80,2}*640
   4-fold quotients : {2,20,6}*480a
   5-fold quotients : {2,16,6}*384
   6-fold quotients : {2,40,2}*320
   8-fold quotients : {2,10,6}*240
   10-fold quotients : {2,8,6}*192
   12-fold quotients : {2,20,2}*160
   15-fold quotients : {2,16,2}*128
   20-fold quotients : {2,4,6}*96a
   24-fold quotients : {2,10,2}*80
   30-fold quotients : {2,8,2}*64
   40-fold quotients : {2,2,6}*48
   48-fold quotients : {2,5,2}*40
   60-fold quotients : {2,4,2}*32
   80-fold quotients : {2,2,3}*24
   120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  7)(  5,  6)(  9, 12)( 10, 11)( 14, 17)( 15, 16)( 19, 22)( 20, 21)
( 24, 27)( 25, 26)( 29, 32)( 30, 31)( 33, 48)( 34, 52)( 35, 51)( 36, 50)
( 37, 49)( 38, 53)( 39, 57)( 40, 56)( 41, 55)( 42, 54)( 43, 58)( 44, 62)
( 45, 61)( 46, 60)( 47, 59)( 63, 93)( 64, 97)( 65, 96)( 66, 95)( 67, 94)
( 68, 98)( 69,102)( 70,101)( 71,100)( 72, 99)( 73,103)( 74,107)( 75,106)
( 76,105)( 77,104)( 78,108)( 79,112)( 80,111)( 81,110)( 82,109)( 83,113)
( 84,117)( 85,116)( 86,115)( 87,114)( 88,118)( 89,122)( 90,121)( 91,120)
( 92,119)(123,183)(124,187)(125,186)(126,185)(127,184)(128,188)(129,192)
(130,191)(131,190)(132,189)(133,193)(134,197)(135,196)(136,195)(137,194)
(138,198)(139,202)(140,201)(141,200)(142,199)(143,203)(144,207)(145,206)
(146,205)(147,204)(148,208)(149,212)(150,211)(151,210)(152,209)(153,228)
(154,232)(155,231)(156,230)(157,229)(158,233)(159,237)(160,236)(161,235)
(162,234)(163,238)(164,242)(165,241)(166,240)(167,239)(168,213)(169,217)
(170,216)(171,215)(172,214)(173,218)(174,222)(175,221)(176,220)(177,219)
(178,223)(179,227)(180,226)(181,225)(182,224);;
s2 := (  3,124)(  4,123)(  5,127)(  6,126)(  7,125)(  8,134)(  9,133)( 10,137)
( 11,136)( 12,135)( 13,129)( 14,128)( 15,132)( 16,131)( 17,130)( 18,139)
( 19,138)( 20,142)( 21,141)( 22,140)( 23,149)( 24,148)( 25,152)( 26,151)
( 27,150)( 28,144)( 29,143)( 30,147)( 31,146)( 32,145)( 33,169)( 34,168)
( 35,172)( 36,171)( 37,170)( 38,179)( 39,178)( 40,182)( 41,181)( 42,180)
( 43,174)( 44,173)( 45,177)( 46,176)( 47,175)( 48,154)( 49,153)( 50,157)
( 51,156)( 52,155)( 53,164)( 54,163)( 55,167)( 56,166)( 57,165)( 58,159)
( 59,158)( 60,162)( 61,161)( 62,160)( 63,214)( 64,213)( 65,217)( 66,216)
( 67,215)( 68,224)( 69,223)( 70,227)( 71,226)( 72,225)( 73,219)( 74,218)
( 75,222)( 76,221)( 77,220)( 78,229)( 79,228)( 80,232)( 81,231)( 82,230)
( 83,239)( 84,238)( 85,242)( 86,241)( 87,240)( 88,234)( 89,233)( 90,237)
( 91,236)( 92,235)( 93,184)( 94,183)( 95,187)( 96,186)( 97,185)( 98,194)
( 99,193)(100,197)(101,196)(102,195)(103,189)(104,188)(105,192)(106,191)
(107,190)(108,199)(109,198)(110,202)(111,201)(112,200)(113,209)(114,208)
(115,212)(116,211)(117,210)(118,204)(119,203)(120,207)(121,206)(122,205);;
s3 := (  3,  8)(  4,  9)(  5, 10)(  6, 11)(  7, 12)( 18, 23)( 19, 24)( 20, 25)
( 21, 26)( 22, 27)( 33, 38)( 34, 39)( 35, 40)( 36, 41)( 37, 42)( 48, 53)
( 49, 54)( 50, 55)( 51, 56)( 52, 57)( 63, 68)( 64, 69)( 65, 70)( 66, 71)
( 67, 72)( 78, 83)( 79, 84)( 80, 85)( 81, 86)( 82, 87)( 93, 98)( 94, 99)
( 95,100)( 96,101)( 97,102)(108,113)(109,114)(110,115)(111,116)(112,117)
(123,128)(124,129)(125,130)(126,131)(127,132)(138,143)(139,144)(140,145)
(141,146)(142,147)(153,158)(154,159)(155,160)(156,161)(157,162)(168,173)
(169,174)(170,175)(171,176)(172,177)(183,188)(184,189)(185,190)(186,191)
(187,192)(198,203)(199,204)(200,205)(201,206)(202,207)(213,218)(214,219)
(215,220)(216,221)(217,222)(228,233)(229,234)(230,235)(231,236)(232,237);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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(242)!(1,2);
s1 := Sym(242)!(  4,  7)(  5,  6)(  9, 12)( 10, 11)( 14, 17)( 15, 16)( 19, 22)
( 20, 21)( 24, 27)( 25, 26)( 29, 32)( 30, 31)( 33, 48)( 34, 52)( 35, 51)
( 36, 50)( 37, 49)( 38, 53)( 39, 57)( 40, 56)( 41, 55)( 42, 54)( 43, 58)
( 44, 62)( 45, 61)( 46, 60)( 47, 59)( 63, 93)( 64, 97)( 65, 96)( 66, 95)
( 67, 94)( 68, 98)( 69,102)( 70,101)( 71,100)( 72, 99)( 73,103)( 74,107)
( 75,106)( 76,105)( 77,104)( 78,108)( 79,112)( 80,111)( 81,110)( 82,109)
( 83,113)( 84,117)( 85,116)( 86,115)( 87,114)( 88,118)( 89,122)( 90,121)
( 91,120)( 92,119)(123,183)(124,187)(125,186)(126,185)(127,184)(128,188)
(129,192)(130,191)(131,190)(132,189)(133,193)(134,197)(135,196)(136,195)
(137,194)(138,198)(139,202)(140,201)(141,200)(142,199)(143,203)(144,207)
(145,206)(146,205)(147,204)(148,208)(149,212)(150,211)(151,210)(152,209)
(153,228)(154,232)(155,231)(156,230)(157,229)(158,233)(159,237)(160,236)
(161,235)(162,234)(163,238)(164,242)(165,241)(166,240)(167,239)(168,213)
(169,217)(170,216)(171,215)(172,214)(173,218)(174,222)(175,221)(176,220)
(177,219)(178,223)(179,227)(180,226)(181,225)(182,224);
s2 := Sym(242)!(  3,124)(  4,123)(  5,127)(  6,126)(  7,125)(  8,134)(  9,133)
( 10,137)( 11,136)( 12,135)( 13,129)( 14,128)( 15,132)( 16,131)( 17,130)
( 18,139)( 19,138)( 20,142)( 21,141)( 22,140)( 23,149)( 24,148)( 25,152)
( 26,151)( 27,150)( 28,144)( 29,143)( 30,147)( 31,146)( 32,145)( 33,169)
( 34,168)( 35,172)( 36,171)( 37,170)( 38,179)( 39,178)( 40,182)( 41,181)
( 42,180)( 43,174)( 44,173)( 45,177)( 46,176)( 47,175)( 48,154)( 49,153)
( 50,157)( 51,156)( 52,155)( 53,164)( 54,163)( 55,167)( 56,166)( 57,165)
( 58,159)( 59,158)( 60,162)( 61,161)( 62,160)( 63,214)( 64,213)( 65,217)
( 66,216)( 67,215)( 68,224)( 69,223)( 70,227)( 71,226)( 72,225)( 73,219)
( 74,218)( 75,222)( 76,221)( 77,220)( 78,229)( 79,228)( 80,232)( 81,231)
( 82,230)( 83,239)( 84,238)( 85,242)( 86,241)( 87,240)( 88,234)( 89,233)
( 90,237)( 91,236)( 92,235)( 93,184)( 94,183)( 95,187)( 96,186)( 97,185)
( 98,194)( 99,193)(100,197)(101,196)(102,195)(103,189)(104,188)(105,192)
(106,191)(107,190)(108,199)(109,198)(110,202)(111,201)(112,200)(113,209)
(114,208)(115,212)(116,211)(117,210)(118,204)(119,203)(120,207)(121,206)
(122,205);
s3 := Sym(242)!(  3,  8)(  4,  9)(  5, 10)(  6, 11)(  7, 12)( 18, 23)( 19, 24)
( 20, 25)( 21, 26)( 22, 27)( 33, 38)( 34, 39)( 35, 40)( 36, 41)( 37, 42)
( 48, 53)( 49, 54)( 50, 55)( 51, 56)( 52, 57)( 63, 68)( 64, 69)( 65, 70)
( 66, 71)( 67, 72)( 78, 83)( 79, 84)( 80, 85)( 81, 86)( 82, 87)( 93, 98)
( 94, 99)( 95,100)( 96,101)( 97,102)(108,113)(109,114)(110,115)(111,116)
(112,117)(123,128)(124,129)(125,130)(126,131)(127,132)(138,143)(139,144)
(140,145)(141,146)(142,147)(153,158)(154,159)(155,160)(156,161)(157,162)
(168,173)(169,174)(170,175)(171,176)(172,177)(183,188)(184,189)(185,190)
(186,191)(187,192)(198,203)(199,204)(200,205)(201,206)(202,207)(213,218)
(214,219)(215,220)(216,221)(217,222)(228,233)(229,234)(230,235)(231,236)
(232,237);
poly := sub<Sym(242)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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 >;

to this polytope