Polytope of Type {78,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {78,12}*1872d
if this polytope has a name.
Group : SmallGroup(1872,1036)
Rank : 3
Schlafli Type : {78,12}
Number of vertices, edges, etc : 78, 468, 12
Order of s₀s₁s₂ : 39
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {78,4}*624c
   6-fold quotients : {39,4}*312
   13-fold quotients : {6,12}*144d
   39-fold quotients : {6,4}*48b
   78-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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