Polytope of Type {4,111}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,111}*888
if this polytope has a name.
Group : SmallGroup(888,48)
Rank : 3
Schlafli Type : {4,111}
Number of vertices, edges, etc : 4, 222, 111
Order of s₀s₁s₂ : 111
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,111,2} of size 1776
Vertex Figure Of :
   {2,4,111} of size 1776
Quotients (Maximal Quotients in Boldface) :
   37-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,111}*1776, {4,222}*1776b, {4,222}*1776c
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);;
s1 := (  2,  3)(  5,145)(  6,147)(  7,146)(  8,148)(  9,141)( 10,143)( 11,142)
( 12,144)( 13,137)( 14,139)( 15,138)( 16,140)( 17,133)( 18,135)( 19,134)
( 20,136)( 21,129)( 22,131)( 23,130)( 24,132)( 25,125)( 26,127)( 27,126)
( 28,128)( 29,121)( 30,123)( 31,122)( 32,124)( 33,117)( 34,119)( 35,118)
( 36,120)( 37,113)( 38,115)( 39,114)( 40,116)( 41,109)( 42,111)( 43,110)
( 44,112)( 45,105)( 46,107)( 47,106)( 48,108)( 49,101)( 50,103)( 51,102)
( 52,104)( 53, 97)( 54, 99)( 55, 98)( 56,100)( 57, 93)( 58, 95)( 59, 94)
( 60, 96)( 61, 89)( 62, 91)( 63, 90)( 64, 92)( 65, 85)( 66, 87)( 67, 86)
( 68, 88)( 69, 81)( 70, 83)( 71, 82)( 72, 84)( 73, 77)( 74, 79)( 75, 78)
( 76, 80);;
s2 := (  1,  5)(  2,  8)(  3,  7)(  4,  6)(  9,145)( 10,148)( 11,147)( 12,146)
( 13,141)( 14,144)( 15,143)( 16,142)( 17,137)( 18,140)( 19,139)( 20,138)
( 21,133)( 22,136)( 23,135)( 24,134)( 25,129)( 26,132)( 27,131)( 28,130)
( 29,125)( 30,128)( 31,127)( 32,126)( 33,121)( 34,124)( 35,123)( 36,122)
( 37,117)( 38,120)( 39,119)( 40,118)( 41,113)( 42,116)( 43,115)( 44,114)
( 45,109)( 46,112)( 47,111)( 48,110)( 49,105)( 50,108)( 51,107)( 52,106)
( 53,101)( 54,104)( 55,103)( 56,102)( 57, 97)( 58,100)( 59, 99)( 60, 98)
( 61, 93)( 62, 96)( 63, 95)( 64, 94)( 65, 89)( 66, 92)( 67, 91)( 68, 90)
( 69, 85)( 70, 88)( 71, 87)( 72, 86)( 73, 81)( 74, 84)( 75, 83)( 76, 82)
( 78, 80);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope