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Polytope of Type {74,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {74,4}*592
Also Known As : {74,4|2}. if this polytope has another name.
Group : SmallGroup(592,39)
Rank : 3
Schlafli Type : {74,4}
Number of vertices, edges, etc : 74, 148, 4
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 :
   {74,4,2} of size 1184
Vertex Figure Of :
   {2,74,4} of size 1184
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {74,2}*296
   4-fold quotients : {37,2}*148
   37-fold quotients : {2,4}*16
   74-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {148,4}*1184, {74,8}*1184
   3-fold covers : {74,12}*1776, {222,4}*1776a
Permutation Representation (GAP) :
s0 := (  2, 37)(  3, 36)(  4, 35)(  5, 34)(  6, 33)(  7, 32)(  8, 31)(  9, 30)
( 10, 29)( 11, 28)( 12, 27)( 13, 26)( 14, 25)( 15, 24)( 16, 23)( 17, 22)
( 18, 21)( 19, 20)( 39, 74)( 40, 73)( 41, 72)( 42, 71)( 43, 70)( 44, 69)
( 45, 68)( 46, 67)( 47, 66)( 48, 65)( 49, 64)( 50, 63)( 51, 62)( 52, 61)
( 53, 60)( 54, 59)( 55, 58)( 56, 57)( 76,111)( 77,110)( 78,109)( 79,108)
( 80,107)( 81,106)( 82,105)( 83,104)( 84,103)( 85,102)( 86,101)( 87,100)
( 88, 99)( 89, 98)( 90, 97)( 91, 96)( 92, 95)( 93, 94)(113,148)(114,147)
(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)(122,139)
(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131);;
s1 := (  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)( 75,113)( 76,112)( 77,148)( 78,147)
( 79,146)( 80,145)( 81,144)( 82,143)( 83,142)( 84,141)( 85,140)( 86,139)
( 87,138)( 88,137)( 89,136)( 90,135)( 91,134)( 92,133)( 93,132)( 94,131)
( 95,130)( 96,129)( 97,128)( 98,127)( 99,126)(100,125)(101,124)(102,123)
(103,122)(104,121)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)
(111,114);;
s2 := (  1, 75)(  2, 76)(  3, 77)(  4, 78)(  5, 79)(  6, 80)(  7, 81)(  8, 82)
(  9, 83)( 10, 84)( 11, 85)( 12, 86)( 13, 87)( 14, 88)( 15, 89)( 16, 90)
( 17, 91)( 18, 92)( 19, 93)( 20, 94)( 21, 95)( 22, 96)( 23, 97)( 24, 98)
( 25, 99)( 26,100)( 27,101)( 28,102)( 29,103)( 30,104)( 31,105)( 32,106)
( 33,107)( 34,108)( 35,109)( 36,110)( 37,111)( 38,112)( 39,113)( 40,114)
( 41,115)( 42,116)( 43,117)( 44,118)( 45,119)( 46,120)( 47,121)( 48,122)
( 49,123)( 50,124)( 51,125)( 52,126)( 53,127)( 54,128)( 55,129)( 56,130)
( 57,131)( 58,132)( 59,133)( 60,134)( 61,135)( 62,136)( 63,137)( 64,138)
( 65,139)( 66,140)( 67,141)( 68,142)( 69,143)( 70,144)( 71,145)( 72,146)
( 73,147)( 74,148);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, 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*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(148)!(  2, 37)(  3, 36)(  4, 35)(  5, 34)(  6, 33)(  7, 32)(  8, 31)
(  9, 30)( 10, 29)( 11, 28)( 12, 27)( 13, 26)( 14, 25)( 15, 24)( 16, 23)
( 17, 22)( 18, 21)( 19, 20)( 39, 74)( 40, 73)( 41, 72)( 42, 71)( 43, 70)
( 44, 69)( 45, 68)( 46, 67)( 47, 66)( 48, 65)( 49, 64)( 50, 63)( 51, 62)
( 52, 61)( 53, 60)( 54, 59)( 55, 58)( 56, 57)( 76,111)( 77,110)( 78,109)
( 79,108)( 80,107)( 81,106)( 82,105)( 83,104)( 84,103)( 85,102)( 86,101)
( 87,100)( 88, 99)( 89, 98)( 90, 97)( 91, 96)( 92, 95)( 93, 94)(113,148)
(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)
(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)
(130,131);
s1 := 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)( 75,113)( 76,112)( 77,148)
( 78,147)( 79,146)( 80,145)( 81,144)( 82,143)( 83,142)( 84,141)( 85,140)
( 86,139)( 87,138)( 88,137)( 89,136)( 90,135)( 91,134)( 92,133)( 93,132)
( 94,131)( 95,130)( 96,129)( 97,128)( 98,127)( 99,126)(100,125)(101,124)
(102,123)(103,122)(104,121)(105,120)(106,119)(107,118)(108,117)(109,116)
(110,115)(111,114);
s2 := Sym(148)!(  1, 75)(  2, 76)(  3, 77)(  4, 78)(  5, 79)(  6, 80)(  7, 81)
(  8, 82)(  9, 83)( 10, 84)( 11, 85)( 12, 86)( 13, 87)( 14, 88)( 15, 89)
( 16, 90)( 17, 91)( 18, 92)( 19, 93)( 20, 94)( 21, 95)( 22, 96)( 23, 97)
( 24, 98)( 25, 99)( 26,100)( 27,101)( 28,102)( 29,103)( 30,104)( 31,105)
( 32,106)( 33,107)( 34,108)( 35,109)( 36,110)( 37,111)( 38,112)( 39,113)
( 40,114)( 41,115)( 42,116)( 43,117)( 44,118)( 45,119)( 46,120)( 47,121)
( 48,122)( 49,123)( 50,124)( 51,125)( 52,126)( 53,127)( 54,128)( 55,129)
( 56,130)( 57,131)( 58,132)( 59,133)( 60,134)( 61,135)( 62,136)( 63,137)
( 64,138)( 65,139)( 66,140)( 67,141)( 68,142)( 69,143)( 70,144)( 71,145)
( 72,146)( 73,147)( 74,148);
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, 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*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.
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