Polytope of Type {21,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,4}*336
if this polytope has a name.
Group : SmallGroup(336,215)
Rank : 3
Schlafli Type : {21,4}
Number of vertices, edges, etc : 42, 84, 8
Order of s₀s₁s₂ : 42
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {21,4,2} of size 672
   {21,4,4} of size 1344
Vertex Figure Of :
   {2,21,4} of size 672
   {4,21,4} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {21,4}*168
   4-fold quotients : {21,2}*84
   7-fold quotients : {3,4}*48
   12-fold quotients : {7,2}*28
   14-fold quotients : {3,4}*24
   28-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {21,8}*672, {42,4}*672
   3-fold covers : {63,4}*1008, {21,12}*1008
   4-fold covers : {21,8}*1344, {84,4}*1344b, {42,4}*1344b, {84,4}*1344c, {42,8}*1344b, {42,8}*1344c
   5-fold covers : {105,4}*1680
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope