Polytope of Type {123,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {123,4}*984
if this polytope has a name.
Group : SmallGroup(984,37)
Rank : 3
Schlafli Type : {123,4}
Number of vertices, edges, etc : 123, 246, 4
Order of s₀s₁s₂ : 123
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {123,4,2} of size 1968
Vertex Figure Of :
   {2,123,4} of size 1968
Quotients (Maximal Quotients in Boldface) :
   41-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {123,4}*1968, {246,4}*1968b, {246,4}*1968c
Permutation Representation (GAP) :

s0 := (  2,  3)(  5,161)(  6,163)(  7,162)(  8,164)(  9,157)( 10,159)( 11,158)
( 12,160)( 13,153)( 14,155)( 15,154)( 16,156)( 17,149)( 18,151)( 19,150)
( 20,152)( 21,145)( 22,147)( 23,146)( 24,148)( 25,141)( 26,143)( 27,142)
( 28,144)( 29,137)( 30,139)( 31,138)( 32,140)( 33,133)( 34,135)( 35,134)
( 36,136)( 37,129)( 38,131)( 39,130)( 40,132)( 41,125)( 42,127)( 43,126)
( 44,128)( 45,121)( 46,123)( 47,122)( 48,124)( 49,117)( 50,119)( 51,118)
( 52,120)( 53,113)( 54,115)( 55,114)( 56,116)( 57,109)( 58,111)( 59,110)
( 60,112)( 61,105)( 62,107)( 63,106)( 64,108)( 65,101)( 66,103)( 67,102)
( 68,104)( 69, 97)( 70, 99)( 71, 98)( 72,100)( 73, 93)( 74, 95)( 75, 94)
( 76, 96)( 77, 89)( 78, 91)( 79, 90)( 80, 92)( 81, 85)( 82, 87)( 83, 86)
( 84, 88);;
s1 := (  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9,161)( 10,162)( 11,164)( 12,163)
( 13,157)( 14,158)( 15,160)( 16,159)( 17,153)( 18,154)( 19,156)( 20,155)
( 21,149)( 22,150)( 23,152)( 24,151)( 25,145)( 26,146)( 27,148)( 28,147)
( 29,141)( 30,142)( 31,144)( 32,143)( 33,137)( 34,138)( 35,140)( 36,139)
( 37,133)( 38,134)( 39,136)( 40,135)( 41,129)( 42,130)( 43,132)( 44,131)
( 45,125)( 46,126)( 47,128)( 48,127)( 49,121)( 50,122)( 51,124)( 52,123)
( 53,117)( 54,118)( 55,120)( 56,119)( 57,113)( 58,114)( 59,116)( 60,115)
( 61,109)( 62,110)( 63,112)( 64,111)( 65,105)( 66,106)( 67,108)( 68,107)
( 69,101)( 70,102)( 71,104)( 72,103)( 73, 97)( 74, 98)( 75,100)( 76, 99)
( 77, 93)( 78, 94)( 79, 96)( 80, 95)( 81, 89)( 82, 90)( 83, 92)( 84, 91)
( 87, 88);;
s2 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(164)!(  2,  3)(  5,161)(  6,163)(  7,162)(  8,164)(  9,157)( 10,159)
( 11,158)( 12,160)( 13,153)( 14,155)( 15,154)( 16,156)( 17,149)( 18,151)
( 19,150)( 20,152)( 21,145)( 22,147)( 23,146)( 24,148)( 25,141)( 26,143)
( 27,142)( 28,144)( 29,137)( 30,139)( 31,138)( 32,140)( 33,133)( 34,135)
( 35,134)( 36,136)( 37,129)( 38,131)( 39,130)( 40,132)( 41,125)( 42,127)
( 43,126)( 44,128)( 45,121)( 46,123)( 47,122)( 48,124)( 49,117)( 50,119)
( 51,118)( 52,120)( 53,113)( 54,115)( 55,114)( 56,116)( 57,109)( 58,111)
( 59,110)( 60,112)( 61,105)( 62,107)( 63,106)( 64,108)( 65,101)( 66,103)
( 67,102)( 68,104)( 69, 97)( 70, 99)( 71, 98)( 72,100)( 73, 93)( 74, 95)
( 75, 94)( 76, 96)( 77, 89)( 78, 91)( 79, 90)( 80, 92)( 81, 85)( 82, 87)
( 83, 86)( 84, 88);
s1 := Sym(164)!(  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9,161)( 10,162)( 11,164)
( 12,163)( 13,157)( 14,158)( 15,160)( 16,159)( 17,153)( 18,154)( 19,156)
( 20,155)( 21,149)( 22,150)( 23,152)( 24,151)( 25,145)( 26,146)( 27,148)
( 28,147)( 29,141)( 30,142)( 31,144)( 32,143)( 33,137)( 34,138)( 35,140)
( 36,139)( 37,133)( 38,134)( 39,136)( 40,135)( 41,129)( 42,130)( 43,132)
( 44,131)( 45,125)( 46,126)( 47,128)( 48,127)( 49,121)( 50,122)( 51,124)
( 52,123)( 53,117)( 54,118)( 55,120)( 56,119)( 57,113)( 58,114)( 59,116)
( 60,115)( 61,109)( 62,110)( 63,112)( 64,111)( 65,105)( 66,106)( 67,108)
( 68,107)( 69,101)( 70,102)( 71,104)( 72,103)( 73, 97)( 74, 98)( 75,100)
( 76, 99)( 77, 93)( 78, 94)( 79, 96)( 80, 95)( 81, 89)( 82, 90)( 83, 92)
( 84, 91)( 87, 88);
s2 := Sym(164)!(  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163);
poly := sub<Sym(164)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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