Polytope of Type {2,4,21}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,21}*672
if this polytope has a name.
Group : SmallGroup(672,1263)
Rank : 4
Schlafli Type : {2,4,21}
Number of vertices, edges, etc : 2, 8, 84, 42
Order of s0s1s2s3 : 42
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,21,2} of size 1344
Vertex Figure Of :
{2,2,4,21} of size 1344
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,21}*336
4-fold quotients : {2,2,21}*168
7-fold quotients : {2,4,3}*96
12-fold quotients : {2,2,7}*56
14-fold quotients : {2,4,3}*48
28-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4,21}*1344b, {2,8,21}*1344, {2,4,42}*1344
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 89)( 4, 90)( 5, 87)( 6, 88)( 7, 93)( 8, 94)( 9, 91)( 10, 92)
( 11, 97)( 12, 98)( 13, 95)( 14, 96)( 15,101)( 16,102)( 17, 99)( 18,100)
( 19,105)( 20,106)( 21,103)( 22,104)( 23,109)( 24,110)( 25,107)( 26,108)
( 27,113)( 28,114)( 29,111)( 30,112)( 31,117)( 32,118)( 33,115)( 34,116)
( 35,121)( 36,122)( 37,119)( 38,120)( 39,125)( 40,126)( 41,123)( 42,124)
( 43,129)( 44,130)( 45,127)( 46,128)( 47,133)( 48,134)( 49,131)( 50,132)
( 51,137)( 52,138)( 53,135)( 54,136)( 55,141)( 56,142)( 57,139)( 58,140)
( 59,145)( 60,146)( 61,143)( 62,144)( 63,149)( 64,150)( 65,147)( 66,148)
( 67,153)( 68,154)( 69,151)( 70,152)( 71,157)( 72,158)( 73,155)( 74,156)
( 75,161)( 76,162)( 77,159)( 78,160)( 79,165)( 80,166)( 81,163)( 82,164)
( 83,169)( 84,170)( 85,167)( 86,168);;
s2 := ( 4, 5)( 7, 27)( 8, 29)( 9, 28)( 10, 30)( 11, 23)( 12, 25)( 13, 24)
( 14, 26)( 15, 19)( 16, 21)( 17, 20)( 18, 22)( 31, 59)( 32, 61)( 33, 60)
( 34, 62)( 35, 83)( 36, 85)( 37, 84)( 38, 86)( 39, 79)( 40, 81)( 41, 80)
( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 71)( 48, 73)( 49, 72)
( 50, 74)( 51, 67)( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)( 57, 64)
( 58, 66)( 88, 89)( 91,111)( 92,113)( 93,112)( 94,114)( 95,107)( 96,109)
( 97,108)( 98,110)( 99,103)(100,105)(101,104)(102,106)(115,143)(116,145)
(117,144)(118,146)(119,167)(120,169)(121,168)(122,170)(123,163)(124,165)
(125,164)(126,166)(127,159)(128,161)(129,160)(130,162)(131,155)(132,157)
(133,156)(134,158)(135,151)(136,153)(137,152)(138,154)(139,147)(140,149)
(141,148)(142,150);;
s3 := ( 3, 63)( 4, 66)( 5, 65)( 6, 64)( 7, 59)( 8, 62)( 9, 61)( 10, 60)
( 11, 83)( 12, 86)( 13, 85)( 14, 84)( 15, 79)( 16, 82)( 17, 81)( 18, 80)
( 19, 75)( 20, 78)( 21, 77)( 22, 76)( 23, 71)( 24, 74)( 25, 73)( 26, 72)
( 27, 67)( 28, 70)( 29, 69)( 30, 68)( 31, 35)( 32, 38)( 33, 37)( 34, 36)
( 39, 55)( 40, 58)( 41, 57)( 42, 56)( 43, 51)( 44, 54)( 45, 53)( 46, 52)
( 48, 50)( 87,147)( 88,150)( 89,149)( 90,148)( 91,143)( 92,146)( 93,145)
( 94,144)( 95,167)( 96,170)( 97,169)( 98,168)( 99,163)(100,166)(101,165)
(102,164)(103,159)(104,162)(105,161)(106,160)(107,155)(108,158)(109,157)
(110,156)(111,151)(112,154)(113,153)(114,152)(115,119)(116,122)(117,121)
(118,120)(123,139)(124,142)(125,141)(126,140)(127,135)(128,138)(129,137)
(130,136)(132,134);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(170)!(1,2);
s1 := Sym(170)!( 3, 89)( 4, 90)( 5, 87)( 6, 88)( 7, 93)( 8, 94)( 9, 91)
( 10, 92)( 11, 97)( 12, 98)( 13, 95)( 14, 96)( 15,101)( 16,102)( 17, 99)
( 18,100)( 19,105)( 20,106)( 21,103)( 22,104)( 23,109)( 24,110)( 25,107)
( 26,108)( 27,113)( 28,114)( 29,111)( 30,112)( 31,117)( 32,118)( 33,115)
( 34,116)( 35,121)( 36,122)( 37,119)( 38,120)( 39,125)( 40,126)( 41,123)
( 42,124)( 43,129)( 44,130)( 45,127)( 46,128)( 47,133)( 48,134)( 49,131)
( 50,132)( 51,137)( 52,138)( 53,135)( 54,136)( 55,141)( 56,142)( 57,139)
( 58,140)( 59,145)( 60,146)( 61,143)( 62,144)( 63,149)( 64,150)( 65,147)
( 66,148)( 67,153)( 68,154)( 69,151)( 70,152)( 71,157)( 72,158)( 73,155)
( 74,156)( 75,161)( 76,162)( 77,159)( 78,160)( 79,165)( 80,166)( 81,163)
( 82,164)( 83,169)( 84,170)( 85,167)( 86,168);
s2 := Sym(170)!( 4, 5)( 7, 27)( 8, 29)( 9, 28)( 10, 30)( 11, 23)( 12, 25)
( 13, 24)( 14, 26)( 15, 19)( 16, 21)( 17, 20)( 18, 22)( 31, 59)( 32, 61)
( 33, 60)( 34, 62)( 35, 83)( 36, 85)( 37, 84)( 38, 86)( 39, 79)( 40, 81)
( 41, 80)( 42, 82)( 43, 75)( 44, 77)( 45, 76)( 46, 78)( 47, 71)( 48, 73)
( 49, 72)( 50, 74)( 51, 67)( 52, 69)( 53, 68)( 54, 70)( 55, 63)( 56, 65)
( 57, 64)( 58, 66)( 88, 89)( 91,111)( 92,113)( 93,112)( 94,114)( 95,107)
( 96,109)( 97,108)( 98,110)( 99,103)(100,105)(101,104)(102,106)(115,143)
(116,145)(117,144)(118,146)(119,167)(120,169)(121,168)(122,170)(123,163)
(124,165)(125,164)(126,166)(127,159)(128,161)(129,160)(130,162)(131,155)
(132,157)(133,156)(134,158)(135,151)(136,153)(137,152)(138,154)(139,147)
(140,149)(141,148)(142,150);
s3 := Sym(170)!( 3, 63)( 4, 66)( 5, 65)( 6, 64)( 7, 59)( 8, 62)( 9, 61)
( 10, 60)( 11, 83)( 12, 86)( 13, 85)( 14, 84)( 15, 79)( 16, 82)( 17, 81)
( 18, 80)( 19, 75)( 20, 78)( 21, 77)( 22, 76)( 23, 71)( 24, 74)( 25, 73)
( 26, 72)( 27, 67)( 28, 70)( 29, 69)( 30, 68)( 31, 35)( 32, 38)( 33, 37)
( 34, 36)( 39, 55)( 40, 58)( 41, 57)( 42, 56)( 43, 51)( 44, 54)( 45, 53)
( 46, 52)( 48, 50)( 87,147)( 88,150)( 89,149)( 90,148)( 91,143)( 92,146)
( 93,145)( 94,144)( 95,167)( 96,170)( 97,169)( 98,168)( 99,163)(100,166)
(101,165)(102,164)(103,159)(104,162)(105,161)(106,160)(107,155)(108,158)
(109,157)(110,156)(111,151)(112,154)(113,153)(114,152)(115,119)(116,122)
(117,121)(118,120)(123,139)(124,142)(125,141)(126,140)(127,135)(128,138)
(129,137)(130,136)(132,134);
poly := sub<Sym(170)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope