Polytope of Type {78,4}

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

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

Permutation Representation (Magma) :

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

References : None.
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