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Polytope of Type {4,74}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,74}*592
Also Known As : {4,74|2}. if this polytope has another name.
Group : SmallGroup(592,39)
Rank : 3
Schlafli Type : {4,74}
Number of vertices, edges, etc : 4, 148, 74
Order of s0s1s2 : 148
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,74,2} of size 1184
Vertex Figure Of :
   {2,4,74} of size 1184
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,74}*296
   4-fold quotients : {2,37}*148
   37-fold quotients : {4,2}*16
   74-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,148}*1184, {8,74}*1184
   3-fold covers : {12,74}*1776, {4,222}*1776a
Permutation Representation (GAP) :
s0 := ( 75,112)( 76,113)( 77,114)( 78,115)( 79,116)( 80,117)( 81,118)( 82,119)
( 83,120)( 84,121)( 85,122)( 86,123)( 87,124)( 88,125)( 89,126)( 90,127)
( 91,128)( 92,129)( 93,130)( 94,131)( 95,132)( 96,133)( 97,134)( 98,135)
( 99,136)(100,137)(101,138)(102,139)(103,140)(104,141)(105,142)(106,143)
(107,144)(108,145)(109,146)(110,147)(111,148);;
s1 := (  1, 75)(  2,111)(  3,110)(  4,109)(  5,108)(  6,107)(  7,106)(  8,105)
(  9,104)( 10,103)( 11,102)( 12,101)( 13,100)( 14, 99)( 15, 98)( 16, 97)
( 17, 96)( 18, 95)( 19, 94)( 20, 93)( 21, 92)( 22, 91)( 23, 90)( 24, 89)
( 25, 88)( 26, 87)( 27, 86)( 28, 85)( 29, 84)( 30, 83)( 31, 82)( 32, 81)
( 33, 80)( 34, 79)( 35, 78)( 36, 77)( 37, 76)( 38,112)( 39,148)( 40,147)
( 41,146)( 42,145)( 43,144)( 44,143)( 45,142)( 46,141)( 47,140)( 48,139)
( 49,138)( 50,137)( 51,136)( 52,135)( 53,134)( 54,133)( 55,132)( 56,131)
( 57,130)( 58,129)( 59,128)( 60,127)( 61,126)( 62,125)( 63,124)( 64,123)
( 65,122)( 66,121)( 67,120)( 68,119)( 69,118)( 70,117)( 71,116)( 72,115)
( 73,114)( 74,113);;
s2 := (  1,  2)(  3, 37)(  4, 36)(  5, 35)(  6, 34)(  7, 33)(  8, 32)(  9, 31)
( 10, 30)( 11, 29)( 12, 28)( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)
( 18, 22)( 19, 21)( 38, 39)( 40, 74)( 41, 73)( 42, 72)( 43, 71)( 44, 70)
( 45, 69)( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 64)( 51, 63)( 52, 62)
( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 75, 76)( 77,111)( 78,110)( 79,109)
( 80,108)( 81,107)( 82,106)( 83,105)( 84,104)( 85,103)( 86,102)( 87,101)
( 88,100)( 89, 99)( 90, 98)( 91, 97)( 92, 96)( 93, 95)(112,113)(114,148)
(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)(121,141)(122,140)
(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132);;
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*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(148)!( 75,112)( 76,113)( 77,114)( 78,115)( 79,116)( 80,117)( 81,118)
( 82,119)( 83,120)( 84,121)( 85,122)( 86,123)( 87,124)( 88,125)( 89,126)
( 90,127)( 91,128)( 92,129)( 93,130)( 94,131)( 95,132)( 96,133)( 97,134)
( 98,135)( 99,136)(100,137)(101,138)(102,139)(103,140)(104,141)(105,142)
(106,143)(107,144)(108,145)(109,146)(110,147)(111,148);
s1 := Sym(148)!(  1, 75)(  2,111)(  3,110)(  4,109)(  5,108)(  6,107)(  7,106)
(  8,105)(  9,104)( 10,103)( 11,102)( 12,101)( 13,100)( 14, 99)( 15, 98)
( 16, 97)( 17, 96)( 18, 95)( 19, 94)( 20, 93)( 21, 92)( 22, 91)( 23, 90)
( 24, 89)( 25, 88)( 26, 87)( 27, 86)( 28, 85)( 29, 84)( 30, 83)( 31, 82)
( 32, 81)( 33, 80)( 34, 79)( 35, 78)( 36, 77)( 37, 76)( 38,112)( 39,148)
( 40,147)( 41,146)( 42,145)( 43,144)( 44,143)( 45,142)( 46,141)( 47,140)
( 48,139)( 49,138)( 50,137)( 51,136)( 52,135)( 53,134)( 54,133)( 55,132)
( 56,131)( 57,130)( 58,129)( 59,128)( 60,127)( 61,126)( 62,125)( 63,124)
( 64,123)( 65,122)( 66,121)( 67,120)( 68,119)( 69,118)( 70,117)( 71,116)
( 72,115)( 73,114)( 74,113);
s2 := Sym(148)!(  1,  2)(  3, 37)(  4, 36)(  5, 35)(  6, 34)(  7, 33)(  8, 32)
(  9, 31)( 10, 30)( 11, 29)( 12, 28)( 13, 27)( 14, 26)( 15, 25)( 16, 24)
( 17, 23)( 18, 22)( 19, 21)( 38, 39)( 40, 74)( 41, 73)( 42, 72)( 43, 71)
( 44, 70)( 45, 69)( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 64)( 51, 63)
( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 75, 76)( 77,111)( 78,110)
( 79,109)( 80,108)( 81,107)( 82,106)( 83,105)( 84,104)( 85,103)( 86,102)
( 87,101)( 88,100)( 89, 99)( 90, 98)( 91, 97)( 92, 96)( 93, 95)(112,113)
(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)(121,141)
(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)
(130,132);
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*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 >; 
 
References : None.
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