Polytope of Type {2,78,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,78,4}*1248a
if this polytope has a name.
Group : SmallGroup(1248,1416)
Rank : 4
Schlafli Type : {2,78,4}
Number of vertices, edges, etc : 2, 78, 156, 4
Order of s₀s₁s₂s₃ : 156
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,78,2}*624
   3-fold quotients : {2,26,4}*416
   4-fold quotients : {2,39,2}*312
   6-fold quotients : {2,26,2}*208
   12-fold quotients : {2,13,2}*104
   13-fold quotients : {2,6,4}*96a
   26-fold quotients : {2,6,2}*48
   39-fold quotients : {2,2,4}*32
   52-fold quotients : {2,3,2}*24
   78-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)( 16, 29)( 17, 41)
( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 22, 36)( 23, 35)( 24, 34)( 25, 33)
( 26, 32)( 27, 31)( 28, 30)( 43, 54)( 44, 53)( 45, 52)( 46, 51)( 47, 50)
( 48, 49)( 55, 68)( 56, 80)( 57, 79)( 58, 78)( 59, 77)( 60, 76)( 61, 75)
( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 82, 93)( 83, 92)
( 84, 91)( 85, 90)( 86, 89)( 87, 88)( 94,107)( 95,119)( 96,118)( 97,117)
( 98,116)( 99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)
(106,108)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,146)
(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152)(141,151)
(142,150)(143,149)(144,148)(145,147);;
s2 := (  3, 17)(  4, 16)(  5, 28)(  6, 27)(  7, 26)(  8, 25)(  9, 24)( 10, 23)
( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 18)( 29, 30)( 31, 41)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 42, 56)( 43, 55)( 44, 67)( 45, 66)( 46, 65)
( 47, 64)( 48, 63)( 49, 62)( 50, 61)( 51, 60)( 52, 59)( 53, 58)( 54, 57)
( 68, 69)( 70, 80)( 71, 79)( 72, 78)( 73, 77)( 74, 76)( 81,134)( 82,133)
( 83,145)( 84,144)( 85,143)( 86,142)( 87,141)( 88,140)( 89,139)( 90,138)
( 91,137)( 92,136)( 93,135)( 94,121)( 95,120)( 96,132)( 97,131)( 98,130)
( 99,129)(100,128)(101,127)(102,126)(103,125)(104,124)(105,123)(106,122)
(107,147)(108,146)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)
(115,152)(116,151)(117,150)(118,149)(119,148);;
s3 := (  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)( 79,157)( 80,158);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(158)!(1,2);
s1 := Sym(158)!(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)( 16, 29)
( 17, 41)( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 22, 36)( 23, 35)( 24, 34)
( 25, 33)( 26, 32)( 27, 31)( 28, 30)( 43, 54)( 44, 53)( 45, 52)( 46, 51)
( 47, 50)( 48, 49)( 55, 68)( 56, 80)( 57, 79)( 58, 78)( 59, 77)( 60, 76)
( 61, 75)( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 82, 93)
( 83, 92)( 84, 91)( 85, 90)( 86, 89)( 87, 88)( 94,107)( 95,119)( 96,118)
( 97,117)( 98,116)( 99,115)(100,114)(101,113)(102,112)(103,111)(104,110)
(105,109)(106,108)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)
(133,146)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152)
(141,151)(142,150)(143,149)(144,148)(145,147);
s2 := Sym(158)!(  3, 17)(  4, 16)(  5, 28)(  6, 27)(  7, 26)(  8, 25)(  9, 24)
( 10, 23)( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 18)( 29, 30)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 56)( 43, 55)( 44, 67)( 45, 66)
( 46, 65)( 47, 64)( 48, 63)( 49, 62)( 50, 61)( 51, 60)( 52, 59)( 53, 58)
( 54, 57)( 68, 69)( 70, 80)( 71, 79)( 72, 78)( 73, 77)( 74, 76)( 81,134)
( 82,133)( 83,145)( 84,144)( 85,143)( 86,142)( 87,141)( 88,140)( 89,139)
( 90,138)( 91,137)( 92,136)( 93,135)( 94,121)( 95,120)( 96,132)( 97,131)
( 98,130)( 99,129)(100,128)(101,127)(102,126)(103,125)(104,124)(105,123)
(106,122)(107,147)(108,146)(109,158)(110,157)(111,156)(112,155)(113,154)
(114,153)(115,152)(116,151)(117,150)(118,149)(119,148);
s3 := Sym(158)!(  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)( 79,157)( 80,158);
poly := sub<Sym(158)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope