Polytope of Type {2,2,4,21}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,4,21}*1344
if this polytope has a name.
Group : SmallGroup(1344,11701)
Rank : 5
Schlafli Type : {2,2,4,21}
Number of vertices, edges, etc : 2, 2, 8, 84, 42
Order of s0s1s2s3s4 : 42
Order of s0s1s2s3s4s3s2s1 : 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 : {2,2,4,21}*672
4-fold quotients : {2,2,2,21}*336
7-fold quotients : {2,2,4,3}*192
12-fold quotients : {2,2,2,7}*112
14-fold quotients : {2,2,4,3}*96
28-fold quotients : {2,2,2,3}*48
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, 65)( 6, 68)( 7, 67)( 8, 66)( 9, 61)( 10, 64)( 11, 63)( 12, 62)
( 13, 85)( 14, 88)( 15, 87)( 16, 86)( 17, 81)( 18, 84)( 19, 83)( 20, 82)
( 21, 77)( 22, 80)( 23, 79)( 24, 78)( 25, 73)( 26, 76)( 27, 75)( 28, 74)
( 29, 69)( 30, 72)( 31, 71)( 32, 70)( 33, 37)( 34, 40)( 35, 39)( 36, 38)
( 41, 57)( 42, 60)( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)( 48, 54)
( 50, 52)( 89,149)( 90,152)( 91,151)( 92,150)( 93,145)( 94,148)( 95,147)
( 96,146)( 97,169)( 98,172)( 99,171)(100,170)(101,165)(102,168)(103,167)
(104,166)(105,161)(106,164)(107,163)(108,162)(109,157)(110,160)(111,159)
(112,158)(113,153)(114,156)(115,155)(116,154)(117,121)(118,124)(119,123)
(120,122)(125,141)(126,144)(127,143)(128,142)(129,137)(130,140)(131,139)
(132,138)(134,136);;
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*s4*s3*s4*s3*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, 65)( 6, 68)( 7, 67)( 8, 66)( 9, 61)( 10, 64)( 11, 63)
( 12, 62)( 13, 85)( 14, 88)( 15, 87)( 16, 86)( 17, 81)( 18, 84)( 19, 83)
( 20, 82)( 21, 77)( 22, 80)( 23, 79)( 24, 78)( 25, 73)( 26, 76)( 27, 75)
( 28, 74)( 29, 69)( 30, 72)( 31, 71)( 32, 70)( 33, 37)( 34, 40)( 35, 39)
( 36, 38)( 41, 57)( 42, 60)( 43, 59)( 44, 58)( 45, 53)( 46, 56)( 47, 55)
( 48, 54)( 50, 52)( 89,149)( 90,152)( 91,151)( 92,150)( 93,145)( 94,148)
( 95,147)( 96,146)( 97,169)( 98,172)( 99,171)(100,170)(101,165)(102,168)
(103,167)(104,166)(105,161)(106,164)(107,163)(108,162)(109,157)(110,160)
(111,159)(112,158)(113,153)(114,156)(115,155)(116,154)(117,121)(118,124)
(119,123)(120,122)(125,141)(126,144)(127,143)(128,142)(129,137)(130,140)
(131,139)(132,138)(134,136);
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*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope