Polytope of Type {2,2,4,42}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,4,42}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11701)
Rank : 5
Schlafli Type : {2,2,4,42}
Number of vertices, edges, etc : 2, 2, 4, 84, 42
Order of s0s1s2s3s4 : 42
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,2,4,21}*672
7-fold quotients : {2,2,4,6}*192b
14-fold quotients : {2,2,4,3}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 5, 91)( 6, 92)( 7, 89)( 8, 90)( 9, 95)( 10, 96)( 11, 93)( 12, 94)
( 13, 99)( 14,100)( 15, 97)( 16, 98)( 17,103)( 18,104)( 19,101)( 20,102)
( 21,107)( 22,108)( 23,105)( 24,106)( 25,111)( 26,112)( 27,109)( 28,110)
( 29,115)( 30,116)( 31,113)( 32,114)( 33,119)( 34,120)( 35,117)( 36,118)
( 37,123)( 38,124)( 39,121)( 40,122)( 41,127)( 42,128)( 43,125)( 44,126)
( 45,131)( 46,132)( 47,129)( 48,130)( 49,135)( 50,136)( 51,133)( 52,134)
( 53,139)( 54,140)( 55,137)( 56,138)( 57,143)( 58,144)( 59,141)( 60,142)
( 61,147)( 62,148)( 63,145)( 64,146)( 65,151)( 66,152)( 67,149)( 68,150)
( 69,155)( 70,156)( 71,153)( 72,154)( 73,159)( 74,160)( 75,157)( 76,158)
( 77,163)( 78,164)( 79,161)( 80,162)( 81,167)( 82,168)( 83,165)( 84,166)
( 85,171)( 86,172)( 87,169)( 88,170);;
s3 := ( 6, 7)( 9, 29)( 10, 31)( 11, 30)( 12, 32)( 13, 25)( 14, 27)( 15, 26)
( 16, 28)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 33, 61)( 34, 63)( 35, 62)
( 36, 64)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 81)( 42, 83)( 43, 82)
( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 73)( 50, 75)( 51, 74)
( 52, 76)( 53, 69)( 54, 71)( 55, 70)( 56, 72)( 57, 65)( 58, 67)( 59, 66)
( 60, 68)( 90, 91)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)( 98,111)
( 99,110)(100,112)(101,105)(102,107)(103,106)(104,108)(117,145)(118,147)
(119,146)(120,148)(121,169)(122,171)(123,170)(124,172)(125,165)(126,167)
(127,166)(128,168)(129,161)(130,163)(131,162)(132,164)(133,157)(134,159)
(135,158)(136,160)(137,153)(138,155)(139,154)(140,156)(141,149)(142,151)
(143,150)(144,152);;
s4 := ( 5,149)( 6,152)( 7,151)( 8,150)( 9,145)( 10,148)( 11,147)( 12,146)
( 13,169)( 14,172)( 15,171)( 16,170)( 17,165)( 18,168)( 19,167)( 20,166)
( 21,161)( 22,164)( 23,163)( 24,162)( 25,157)( 26,160)( 27,159)( 28,158)
( 29,153)( 30,156)( 31,155)( 32,154)( 33,121)( 34,124)( 35,123)( 36,122)
( 37,117)( 38,120)( 39,119)( 40,118)( 41,141)( 42,144)( 43,143)( 44,142)
( 45,137)( 46,140)( 47,139)( 48,138)( 49,133)( 50,136)( 51,135)( 52,134)
( 53,129)( 54,132)( 55,131)( 56,130)( 57,125)( 58,128)( 59,127)( 60,126)
( 61, 93)( 62, 96)( 63, 95)( 64, 94)( 65, 89)( 66, 92)( 67, 91)( 68, 90)
( 69,113)( 70,116)( 71,115)( 72,114)( 73,109)( 74,112)( 75,111)( 76,110)
( 77,105)( 78,108)( 79,107)( 80,106)( 81,101)( 82,104)( 83,103)( 84,102)
( 85, 97)( 86,100)( 87, 99)( 88, 98);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(172)!(1,2);
s1 := Sym(172)!(3,4);
s2 := Sym(172)!( 5, 91)( 6, 92)( 7, 89)( 8, 90)( 9, 95)( 10, 96)( 11, 93)
( 12, 94)( 13, 99)( 14,100)( 15, 97)( 16, 98)( 17,103)( 18,104)( 19,101)
( 20,102)( 21,107)( 22,108)( 23,105)( 24,106)( 25,111)( 26,112)( 27,109)
( 28,110)( 29,115)( 30,116)( 31,113)( 32,114)( 33,119)( 34,120)( 35,117)
( 36,118)( 37,123)( 38,124)( 39,121)( 40,122)( 41,127)( 42,128)( 43,125)
( 44,126)( 45,131)( 46,132)( 47,129)( 48,130)( 49,135)( 50,136)( 51,133)
( 52,134)( 53,139)( 54,140)( 55,137)( 56,138)( 57,143)( 58,144)( 59,141)
( 60,142)( 61,147)( 62,148)( 63,145)( 64,146)( 65,151)( 66,152)( 67,149)
( 68,150)( 69,155)( 70,156)( 71,153)( 72,154)( 73,159)( 74,160)( 75,157)
( 76,158)( 77,163)( 78,164)( 79,161)( 80,162)( 81,167)( 82,168)( 83,165)
( 84,166)( 85,171)( 86,172)( 87,169)( 88,170);
s3 := Sym(172)!( 6, 7)( 9, 29)( 10, 31)( 11, 30)( 12, 32)( 13, 25)( 14, 27)
( 15, 26)( 16, 28)( 17, 21)( 18, 23)( 19, 22)( 20, 24)( 33, 61)( 34, 63)
( 35, 62)( 36, 64)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 81)( 42, 83)
( 43, 82)( 44, 84)( 45, 77)( 46, 79)( 47, 78)( 48, 80)( 49, 73)( 50, 75)
( 51, 74)( 52, 76)( 53, 69)( 54, 71)( 55, 70)( 56, 72)( 57, 65)( 58, 67)
( 59, 66)( 60, 68)( 90, 91)( 93,113)( 94,115)( 95,114)( 96,116)( 97,109)
( 98,111)( 99,110)(100,112)(101,105)(102,107)(103,106)(104,108)(117,145)
(118,147)(119,146)(120,148)(121,169)(122,171)(123,170)(124,172)(125,165)
(126,167)(127,166)(128,168)(129,161)(130,163)(131,162)(132,164)(133,157)
(134,159)(135,158)(136,160)(137,153)(138,155)(139,154)(140,156)(141,149)
(142,151)(143,150)(144,152);
s4 := Sym(172)!( 5,149)( 6,152)( 7,151)( 8,150)( 9,145)( 10,148)( 11,147)
( 12,146)( 13,169)( 14,172)( 15,171)( 16,170)( 17,165)( 18,168)( 19,167)
( 20,166)( 21,161)( 22,164)( 23,163)( 24,162)( 25,157)( 26,160)( 27,159)
( 28,158)( 29,153)( 30,156)( 31,155)( 32,154)( 33,121)( 34,124)( 35,123)
( 36,122)( 37,117)( 38,120)( 39,119)( 40,118)( 41,141)( 42,144)( 43,143)
( 44,142)( 45,137)( 46,140)( 47,139)( 48,138)( 49,133)( 50,136)( 51,135)
( 52,134)( 53,129)( 54,132)( 55,131)( 56,130)( 57,125)( 58,128)( 59,127)
( 60,126)( 61, 93)( 62, 96)( 63, 95)( 64, 94)( 65, 89)( 66, 92)( 67, 91)
( 68, 90)( 69,113)( 70,116)( 71,115)( 72,114)( 73,109)( 74,112)( 75,111)
( 76,110)( 77,105)( 78,108)( 79,107)( 80,106)( 81,101)( 82,104)( 83,103)
( 84,102)( 85, 97)( 86,100)( 87, 99)( 88, 98);
poly := sub<Sym(172)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope