Polytope of Type {6,57}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,57}*684
if this polytope has a name.
Group : SmallGroup(684,41)
Rank : 3
Schlafli Type : {6,57}
Number of vertices, edges, etc : 6, 171, 57
Order of s₀s₁s₂ : 114
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,57,2} of size 1368
Vertex Figure Of :
   {2,6,57} of size 1368
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,57}*228
   9-fold quotients : {2,19}*76
   19-fold quotients : {6,3}*36
   57-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,114}*1368c
Permutation Representation (GAP) :

s0 := ( 58,115)( 59,116)( 60,117)( 61,118)( 62,119)( 63,120)( 64,121)( 65,122)
( 66,123)( 67,124)( 68,125)( 69,126)( 70,127)( 71,128)( 72,129)( 73,130)
( 74,131)( 75,132)( 76,133)( 77,134)( 78,135)( 79,136)( 80,137)( 81,138)
( 82,139)( 83,140)( 84,141)( 85,142)( 86,143)( 87,144)( 88,145)( 89,146)
( 90,147)( 91,148)( 92,149)( 93,150)( 94,151)( 95,152)( 96,153)( 97,154)
( 98,155)( 99,156)(100,157)(101,158)(102,159)(103,160)(104,161)(105,162)
(106,163)(107,164)(108,165)(109,166)(110,167)(111,168)(112,169)(113,170)
(114,171);;
s1 := (  1, 58)(  2, 76)(  3, 75)(  4, 74)(  5, 73)(  6, 72)(  7, 71)(  8, 70)
(  9, 69)( 10, 68)( 11, 67)( 12, 66)( 13, 65)( 14, 64)( 15, 63)( 16, 62)
( 17, 61)( 18, 60)( 19, 59)( 20, 96)( 21,114)( 22,113)( 23,112)( 24,111)
( 25,110)( 26,109)( 27,108)( 28,107)( 29,106)( 30,105)( 31,104)( 32,103)
( 33,102)( 34,101)( 35,100)( 36, 99)( 37, 98)( 38, 97)( 39, 77)( 40, 95)
( 41, 94)( 42, 93)( 43, 92)( 44, 91)( 45, 90)( 46, 89)( 47, 88)( 48, 87)
( 49, 86)( 50, 85)( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)( 56, 79)
( 57, 78)(116,133)(117,132)(118,131)(119,130)(120,129)(121,128)(122,127)
(123,126)(124,125)(134,153)(135,171)(136,170)(137,169)(138,168)(139,167)
(140,166)(141,165)(142,164)(143,163)(144,162)(145,161)(146,160)(147,159)
(148,158)(149,157)(150,156)(151,155)(152,154);;
s2 := (  1, 21)(  2, 20)(  3, 38)(  4, 37)(  5, 36)(  6, 35)(  7, 34)(  8, 33)
(  9, 32)( 10, 31)( 11, 30)( 12, 29)( 13, 28)( 14, 27)( 15, 26)( 16, 25)
( 17, 24)( 18, 23)( 19, 22)( 39, 40)( 41, 57)( 42, 56)( 43, 55)( 44, 54)
( 45, 53)( 46, 52)( 47, 51)( 48, 50)( 58,135)( 59,134)( 60,152)( 61,151)
( 62,150)( 63,149)( 64,148)( 65,147)( 66,146)( 67,145)( 68,144)( 69,143)
( 70,142)( 71,141)( 72,140)( 73,139)( 74,138)( 75,137)( 76,136)( 77,116)
( 78,115)( 79,133)( 80,132)( 81,131)( 82,130)( 83,129)( 84,128)( 85,127)
( 86,126)( 87,125)( 88,124)( 89,123)( 90,122)( 91,121)( 92,120)( 93,119)
( 94,118)( 95,117)( 96,154)( 97,153)( 98,171)( 99,170)(100,169)(101,168)
(102,167)(103,166)(104,165)(105,164)(106,163)(107,162)(108,161)(109,160)
(110,159)(111,158)(112,157)(113,156)(114,155);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(171)!( 58,115)( 59,116)( 60,117)( 61,118)( 62,119)( 63,120)( 64,121)
( 65,122)( 66,123)( 67,124)( 68,125)( 69,126)( 70,127)( 71,128)( 72,129)
( 73,130)( 74,131)( 75,132)( 76,133)( 77,134)( 78,135)( 79,136)( 80,137)
( 81,138)( 82,139)( 83,140)( 84,141)( 85,142)( 86,143)( 87,144)( 88,145)
( 89,146)( 90,147)( 91,148)( 92,149)( 93,150)( 94,151)( 95,152)( 96,153)
( 97,154)( 98,155)( 99,156)(100,157)(101,158)(102,159)(103,160)(104,161)
(105,162)(106,163)(107,164)(108,165)(109,166)(110,167)(111,168)(112,169)
(113,170)(114,171);
s1 := Sym(171)!(  1, 58)(  2, 76)(  3, 75)(  4, 74)(  5, 73)(  6, 72)(  7, 71)
(  8, 70)(  9, 69)( 10, 68)( 11, 67)( 12, 66)( 13, 65)( 14, 64)( 15, 63)
( 16, 62)( 17, 61)( 18, 60)( 19, 59)( 20, 96)( 21,114)( 22,113)( 23,112)
( 24,111)( 25,110)( 26,109)( 27,108)( 28,107)( 29,106)( 30,105)( 31,104)
( 32,103)( 33,102)( 34,101)( 35,100)( 36, 99)( 37, 98)( 38, 97)( 39, 77)
( 40, 95)( 41, 94)( 42, 93)( 43, 92)( 44, 91)( 45, 90)( 46, 89)( 47, 88)
( 48, 87)( 49, 86)( 50, 85)( 51, 84)( 52, 83)( 53, 82)( 54, 81)( 55, 80)
( 56, 79)( 57, 78)(116,133)(117,132)(118,131)(119,130)(120,129)(121,128)
(122,127)(123,126)(124,125)(134,153)(135,171)(136,170)(137,169)(138,168)
(139,167)(140,166)(141,165)(142,164)(143,163)(144,162)(145,161)(146,160)
(147,159)(148,158)(149,157)(150,156)(151,155)(152,154);
s2 := Sym(171)!(  1, 21)(  2, 20)(  3, 38)(  4, 37)(  5, 36)(  6, 35)(  7, 34)
(  8, 33)(  9, 32)( 10, 31)( 11, 30)( 12, 29)( 13, 28)( 14, 27)( 15, 26)
( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 39, 40)( 41, 57)( 42, 56)( 43, 55)
( 44, 54)( 45, 53)( 46, 52)( 47, 51)( 48, 50)( 58,135)( 59,134)( 60,152)
( 61,151)( 62,150)( 63,149)( 64,148)( 65,147)( 66,146)( 67,145)( 68,144)
( 69,143)( 70,142)( 71,141)( 72,140)( 73,139)( 74,138)( 75,137)( 76,136)
( 77,116)( 78,115)( 79,133)( 80,132)( 81,131)( 82,130)( 83,129)( 84,128)
( 85,127)( 86,126)( 87,125)( 88,124)( 89,123)( 90,122)( 91,121)( 92,120)
( 93,119)( 94,118)( 95,117)( 96,154)( 97,153)( 98,171)( 99,170)(100,169)
(101,168)(102,167)(103,166)(104,165)(105,164)(106,163)(107,162)(108,161)
(109,160)(110,159)(111,158)(112,157)(113,156)(114,155);
poly := sub<Sym(171)|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, s2*s0*s1*s0*s1*s2*s0*s1*s0*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 >;

References : None.
to this polytope