Polytope of Type {4,40}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,40}*1280b
if this polytope has a name.
Group : SmallGroup(1280,90280)
Rank : 3
Schlafli Type : {4,40}
Number of vertices, edges, etc : 16, 320, 160
Order of s₀s₁s₂ : 20
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {4,40}*640b
   4-fold quotients : {4,20}*320
   5-fold quotients : {4,8}*256b
   8-fold quotients : {4,20}*160
   10-fold quotients : {4,8}*128b
   16-fold quotients : {2,20}*80, {4,10}*80
   20-fold quotients : {4,4}*64
   32-fold quotients : {2,10}*40
   40-fold quotients : {4,4}*32
   64-fold quotients : {2,5}*20
   80-fold quotients : {2,4}*16, {4,2}*16
   160-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)( 33, 38)
( 34, 39)( 35, 40)( 41, 76)( 42, 77)( 43, 78)( 44, 79)( 45, 80)( 46, 71)
( 47, 72)( 48, 73)( 49, 74)( 50, 75)( 51, 61)( 52, 62)( 53, 63)( 54, 64)
( 55, 65)( 56, 66)( 57, 67)( 58, 68)( 59, 69)( 60, 70)( 91, 96)( 92, 97)
( 93, 98)( 94, 99)( 95,100)(111,116)(112,117)(113,118)(114,119)(115,120)
(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)(128,153)
(129,154)(130,155)(131,141)(132,142)(133,143)(134,144)(135,145)(136,146)
(137,147)(138,148)(139,149)(140,150);;
s1 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 21, 31)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 36)( 27, 40)( 28, 39)
( 29, 38)( 30, 37)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)
( 57, 60)( 58, 59)( 61, 71)( 62, 75)( 63, 74)( 64, 73)( 65, 72)( 66, 76)
( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 81,121)( 82,125)( 83,124)( 84,123)
( 85,122)( 86,126)( 87,130)( 88,129)( 89,128)( 90,127)( 91,131)( 92,135)
( 93,134)( 94,133)( 95,132)( 96,136)( 97,140)( 98,139)( 99,138)(100,137)
(101,151)(102,155)(103,154)(104,153)(105,152)(106,156)(107,160)(108,159)
(109,158)(110,157)(111,141)(112,145)(113,144)(114,143)(115,142)(116,146)
(117,150)(118,149)(119,148)(120,147);;
s2 := (  1, 83)(  2, 82)(  3, 81)(  4, 85)(  5, 84)(  6, 88)(  7, 87)(  8, 86)
(  9, 90)( 10, 89)( 11, 93)( 12, 92)( 13, 91)( 14, 95)( 15, 94)( 16, 98)
( 17, 97)( 18, 96)( 19,100)( 20, 99)( 21,108)( 22,107)( 23,106)( 24,110)
( 25,109)( 26,103)( 27,102)( 28,101)( 29,105)( 30,104)( 31,118)( 32,117)
( 33,116)( 34,120)( 35,119)( 36,113)( 37,112)( 38,111)( 39,115)( 40,114)
( 41,158)( 42,157)( 43,156)( 44,160)( 45,159)( 46,153)( 47,152)( 48,151)
( 49,155)( 50,154)( 51,148)( 52,147)( 53,146)( 54,150)( 55,149)( 56,143)
( 57,142)( 58,141)( 59,145)( 60,144)( 61,138)( 62,137)( 63,136)( 64,140)
( 65,139)( 66,133)( 67,132)( 68,131)( 69,135)( 70,134)( 71,128)( 72,127)
( 73,126)( 74,130)( 75,129)( 76,123)( 77,122)( 78,121)( 79,125)( 80,124);;
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, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*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(160)!( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)
( 33, 38)( 34, 39)( 35, 40)( 41, 76)( 42, 77)( 43, 78)( 44, 79)( 45, 80)
( 46, 71)( 47, 72)( 48, 73)( 49, 74)( 50, 75)( 51, 61)( 52, 62)( 53, 63)
( 54, 64)( 55, 65)( 56, 66)( 57, 67)( 58, 68)( 59, 69)( 60, 70)( 91, 96)
( 92, 97)( 93, 98)( 94, 99)( 95,100)(111,116)(112,117)(113,118)(114,119)
(115,120)(121,156)(122,157)(123,158)(124,159)(125,160)(126,151)(127,152)
(128,153)(129,154)(130,155)(131,141)(132,142)(133,143)(134,144)(135,145)
(136,146)(137,147)(138,148)(139,149)(140,150);
s1 := Sym(160)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 21, 31)( 22, 35)( 23, 34)( 24, 33)( 25, 32)( 26, 36)( 27, 40)
( 28, 39)( 29, 38)( 30, 37)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)
( 53, 54)( 57, 60)( 58, 59)( 61, 71)( 62, 75)( 63, 74)( 64, 73)( 65, 72)
( 66, 76)( 67, 80)( 68, 79)( 69, 78)( 70, 77)( 81,121)( 82,125)( 83,124)
( 84,123)( 85,122)( 86,126)( 87,130)( 88,129)( 89,128)( 90,127)( 91,131)
( 92,135)( 93,134)( 94,133)( 95,132)( 96,136)( 97,140)( 98,139)( 99,138)
(100,137)(101,151)(102,155)(103,154)(104,153)(105,152)(106,156)(107,160)
(108,159)(109,158)(110,157)(111,141)(112,145)(113,144)(114,143)(115,142)
(116,146)(117,150)(118,149)(119,148)(120,147);
s2 := Sym(160)!(  1, 83)(  2, 82)(  3, 81)(  4, 85)(  5, 84)(  6, 88)(  7, 87)
(  8, 86)(  9, 90)( 10, 89)( 11, 93)( 12, 92)( 13, 91)( 14, 95)( 15, 94)
( 16, 98)( 17, 97)( 18, 96)( 19,100)( 20, 99)( 21,108)( 22,107)( 23,106)
( 24,110)( 25,109)( 26,103)( 27,102)( 28,101)( 29,105)( 30,104)( 31,118)
( 32,117)( 33,116)( 34,120)( 35,119)( 36,113)( 37,112)( 38,111)( 39,115)
( 40,114)( 41,158)( 42,157)( 43,156)( 44,160)( 45,159)( 46,153)( 47,152)
( 48,151)( 49,155)( 50,154)( 51,148)( 52,147)( 53,146)( 54,150)( 55,149)
( 56,143)( 57,142)( 58,141)( 59,145)( 60,144)( 61,138)( 62,137)( 63,136)
( 64,140)( 65,139)( 66,133)( 67,132)( 68,131)( 69,135)( 70,134)( 71,128)
( 72,127)( 73,126)( 74,130)( 75,129)( 76,123)( 77,122)( 78,121)( 79,125)
( 80,124);
poly := sub<Sym(160)|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, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
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