Polytope of Type {57,6}

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

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