Polytope of Type {4,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,78}*624a
Also Known As : {4,78|2}. if this polytope has another name.
Group : SmallGroup(624,228)
Rank : 3
Schlafli Type : {4,78}
Number of vertices, edges, etc : 4, 156, 78
Order of s₀s₁s₂ : 156
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,78,2} of size 1248
Vertex Figure Of :
   {2,4,78} of size 1248
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,78}*312
   3-fold quotients : {4,26}*208
   4-fold quotients : {2,39}*156
   6-fold quotients : {2,26}*104
   12-fold quotients : {2,13}*52
   13-fold quotients : {4,6}*48a
   26-fold quotients : {2,6}*24
   39-fold quotients : {4,2}*16
   52-fold quotients : {2,3}*12
   78-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,156}*1248a, {8,78}*1248
   3-fold covers : {4,234}*1872a, {12,78}*1872b, {12,78}*1872c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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