Polytope of Type {4,165}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,165}*1320
if this polytope has a name.
Group : SmallGroup(1320,157)
Rank : 3
Schlafli Type : {4,165}
Number of vertices, edges, etc : 4, 330, 165
Order of s₀s₁s₂ : 165
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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) :
   5-fold quotients : {4,33}*264
   11-fold quotients : {4,15}*120
   55-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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