Polytope of Type {4,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,42}*336b
if this polytope has a name.
Group : SmallGroup(336,215)
Rank : 3
Schlafli Type : {4,42}
Number of vertices, edges, etc : 4, 84, 42
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,42,2} of size 672
   {4,42,4} of size 1344
   {4,42,4} of size 1344
   {4,42,4} of size 1344
Vertex Figure Of :
   {2,4,42} of size 672
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,21}*168
   7-fold quotients : {4,6}*48c
   14-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,84}*672b, {4,84}*672c, {4,42}*672
   3-fold covers : {4,126}*1008b
   4-fold covers : {4,42}*1344a, {4,168}*1344c, {4,168}*1344d, {4,84}*1344b, {4,42}*1344b, {4,84}*1344c, {8,42}*1344b, {8,42}*1344c
   5-fold covers : {4,210}*1680b
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168);;
s1 := (  2,  3)(  5, 25)(  6, 27)(  7, 26)(  8, 28)(  9, 21)( 10, 23)( 11, 22)
( 12, 24)( 13, 17)( 14, 19)( 15, 18)( 16, 20)( 29, 57)( 30, 59)( 31, 58)
( 32, 60)( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 77)( 38, 79)( 39, 78)
( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 69)( 46, 71)( 47, 70)
( 48, 72)( 49, 65)( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)
( 56, 64)( 86, 87)( 89,109)( 90,111)( 91,110)( 92,112)( 93,105)( 94,107)
( 95,106)( 96,108)( 97,101)( 98,103)( 99,102)(100,104)(113,141)(114,143)
(115,142)(116,144)(117,165)(118,167)(119,166)(120,168)(121,161)(122,163)
(123,162)(124,164)(125,157)(126,159)(127,158)(128,160)(129,153)(130,155)
(131,154)(132,156)(133,149)(134,151)(135,150)(136,152)(137,145)(138,147)
(139,146)(140,148);;
s2 := (  1,145)(  2,148)(  3,147)(  4,146)(  5,141)(  6,144)(  7,143)(  8,142)
(  9,165)( 10,168)( 11,167)( 12,166)( 13,161)( 14,164)( 15,163)( 16,162)
( 17,157)( 18,160)( 19,159)( 20,158)( 21,153)( 22,156)( 23,155)( 24,154)
( 25,149)( 26,152)( 27,151)( 28,150)( 29,117)( 30,120)( 31,119)( 32,118)
( 33,113)( 34,116)( 35,115)( 36,114)( 37,137)( 38,140)( 39,139)( 40,138)
( 41,133)( 42,136)( 43,135)( 44,134)( 45,129)( 46,132)( 47,131)( 48,130)
( 49,125)( 50,128)( 51,127)( 52,126)( 53,121)( 54,124)( 55,123)( 56,122)
( 57, 89)( 58, 92)( 59, 91)( 60, 90)( 61, 85)( 62, 88)( 63, 87)( 64, 86)
( 65,109)( 66,112)( 67,111)( 68,110)( 69,105)( 70,108)( 71,107)( 72,106)
( 73,101)( 74,104)( 75,103)( 76,102)( 77, 97)( 78,100)( 79, 99)( 80, 98)
( 81, 93)( 82, 96)( 83, 95)( 84, 94);;
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*s0*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(168)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168);
s1 := Sym(168)!(  2,  3)(  5, 25)(  6, 27)(  7, 26)(  8, 28)(  9, 21)( 10, 23)
( 11, 22)( 12, 24)( 13, 17)( 14, 19)( 15, 18)( 16, 20)( 29, 57)( 30, 59)
( 31, 58)( 32, 60)( 33, 81)( 34, 83)( 35, 82)( 36, 84)( 37, 77)( 38, 79)
( 39, 78)( 40, 80)( 41, 73)( 42, 75)( 43, 74)( 44, 76)( 45, 69)( 46, 71)
( 47, 70)( 48, 72)( 49, 65)( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)
( 55, 62)( 56, 64)( 86, 87)( 89,109)( 90,111)( 91,110)( 92,112)( 93,105)
( 94,107)( 95,106)( 96,108)( 97,101)( 98,103)( 99,102)(100,104)(113,141)
(114,143)(115,142)(116,144)(117,165)(118,167)(119,166)(120,168)(121,161)
(122,163)(123,162)(124,164)(125,157)(126,159)(127,158)(128,160)(129,153)
(130,155)(131,154)(132,156)(133,149)(134,151)(135,150)(136,152)(137,145)
(138,147)(139,146)(140,148);
s2 := Sym(168)!(  1,145)(  2,148)(  3,147)(  4,146)(  5,141)(  6,144)(  7,143)
(  8,142)(  9,165)( 10,168)( 11,167)( 12,166)( 13,161)( 14,164)( 15,163)
( 16,162)( 17,157)( 18,160)( 19,159)( 20,158)( 21,153)( 22,156)( 23,155)
( 24,154)( 25,149)( 26,152)( 27,151)( 28,150)( 29,117)( 30,120)( 31,119)
( 32,118)( 33,113)( 34,116)( 35,115)( 36,114)( 37,137)( 38,140)( 39,139)
( 40,138)( 41,133)( 42,136)( 43,135)( 44,134)( 45,129)( 46,132)( 47,131)
( 48,130)( 49,125)( 50,128)( 51,127)( 52,126)( 53,121)( 54,124)( 55,123)
( 56,122)( 57, 89)( 58, 92)( 59, 91)( 60, 90)( 61, 85)( 62, 88)( 63, 87)
( 64, 86)( 65,109)( 66,112)( 67,111)( 68,110)( 69,105)( 70,108)( 71,107)
( 72,106)( 73,101)( 74,104)( 75,103)( 76,102)( 77, 97)( 78,100)( 79, 99)
( 80, 98)( 81, 93)( 82, 96)( 83, 95)( 84, 94);
poly := sub<Sym(168)|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*s0*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 >;

References : None.
to this polytope