Polytope of Type {6,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,84}*1512a
if this polytope has a name.
Group : SmallGroup(1512,482)
Rank : 3
Schlafli Type : {6,84}
Number of vertices, edges, etc : 9, 378, 126
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,28}*504
   7-fold quotients : {6,12}*216a
   21-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)( 29, 50)
( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)( 37, 58)
( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 64,127)( 65,128)( 66,129)
( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)( 72,135)( 73,136)( 74,137)
( 75,138)( 76,139)( 77,140)( 78,141)( 79,142)( 80,143)( 81,144)( 82,145)
( 83,146)( 84,147)( 85,169)( 86,170)( 87,171)( 88,172)( 89,173)( 90,174)
( 91,175)( 92,176)( 93,177)( 94,178)( 95,179)( 96,180)( 97,181)( 98,182)
( 99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)(106,148)
(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)(113,155)(114,156)
(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,163)(122,164)
(123,165)(124,166)(125,167)(126,168);;
s1 := (  1, 64)(  2, 66)(  3, 65)(  4, 82)(  5, 84)(  6, 83)(  7, 79)(  8, 81)
(  9, 80)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)( 16, 70)
( 17, 72)( 18, 71)( 19, 67)( 20, 69)( 21, 68)( 22, 86)( 23, 85)( 24, 87)
( 25,104)( 26,103)( 27,105)( 28,101)( 29,100)( 30,102)( 31, 98)( 32, 97)
( 33, 99)( 34, 95)( 35, 94)( 36, 96)( 37, 92)( 38, 91)( 39, 93)( 40, 89)
( 41, 88)( 42, 90)( 43,108)( 44,107)( 45,106)( 46,126)( 47,125)( 48,124)
( 49,123)( 50,122)( 51,121)( 52,120)( 53,119)( 54,118)( 55,117)( 56,116)
( 57,115)( 58,114)( 59,113)( 60,112)( 61,111)( 62,110)( 63,109)(128,129)
(130,145)(131,147)(132,146)(133,142)(134,144)(135,143)(136,139)(137,141)
(138,140)(148,149)(151,167)(152,166)(153,168)(154,164)(155,163)(156,165)
(157,161)(158,160)(159,162)(169,171)(172,189)(173,188)(174,187)(175,186)
(176,185)(177,184)(178,183)(179,182)(180,181);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  7, 19)(  8, 21)(  9, 20)( 10, 16)( 11, 18)
( 12, 17)( 14, 15)( 22, 67)( 23, 69)( 24, 68)( 25, 64)( 26, 66)( 27, 65)
( 28, 82)( 29, 84)( 30, 83)( 31, 79)( 32, 81)( 33, 80)( 34, 76)( 35, 78)
( 36, 77)( 37, 73)( 38, 75)( 39, 74)( 40, 70)( 41, 72)( 42, 71)( 43,130)
( 44,132)( 45,131)( 46,127)( 47,129)( 48,128)( 49,145)( 50,147)( 51,146)
( 52,142)( 53,144)( 54,143)( 55,139)( 56,141)( 57,140)( 58,136)( 59,138)
( 60,137)( 61,133)( 62,135)( 63,134)( 85, 89)( 86, 88)( 87, 90)( 91,104)
( 92,103)( 93,105)( 94,101)( 95,100)( 96,102)( 97, 98)(106,153)(107,152)
(108,151)(109,150)(110,149)(111,148)(112,168)(113,167)(114,166)(115,165)
(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)(123,157)
(124,156)(125,155)(126,154)(169,173)(170,172)(171,174)(175,188)(176,187)
(177,189)(178,185)(179,184)(180,186)(181,182);;
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*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(189)!( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)
( 29, 50)( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)
( 37, 58)( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 64,127)( 65,128)
( 66,129)( 67,130)( 68,131)( 69,132)( 70,133)( 71,134)( 72,135)( 73,136)
( 74,137)( 75,138)( 76,139)( 77,140)( 78,141)( 79,142)( 80,143)( 81,144)
( 82,145)( 83,146)( 84,147)( 85,169)( 86,170)( 87,171)( 88,172)( 89,173)
( 90,174)( 91,175)( 92,176)( 93,177)( 94,178)( 95,179)( 96,180)( 97,181)
( 98,182)( 99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)
(106,148)(107,149)(108,150)(109,151)(110,152)(111,153)(112,154)(113,155)
(114,156)(115,157)(116,158)(117,159)(118,160)(119,161)(120,162)(121,163)
(122,164)(123,165)(124,166)(125,167)(126,168);
s1 := Sym(189)!(  1, 64)(  2, 66)(  3, 65)(  4, 82)(  5, 84)(  6, 83)(  7, 79)
(  8, 81)(  9, 80)( 10, 76)( 11, 78)( 12, 77)( 13, 73)( 14, 75)( 15, 74)
( 16, 70)( 17, 72)( 18, 71)( 19, 67)( 20, 69)( 21, 68)( 22, 86)( 23, 85)
( 24, 87)( 25,104)( 26,103)( 27,105)( 28,101)( 29,100)( 30,102)( 31, 98)
( 32, 97)( 33, 99)( 34, 95)( 35, 94)( 36, 96)( 37, 92)( 38, 91)( 39, 93)
( 40, 89)( 41, 88)( 42, 90)( 43,108)( 44,107)( 45,106)( 46,126)( 47,125)
( 48,124)( 49,123)( 50,122)( 51,121)( 52,120)( 53,119)( 54,118)( 55,117)
( 56,116)( 57,115)( 58,114)( 59,113)( 60,112)( 61,111)( 62,110)( 63,109)
(128,129)(130,145)(131,147)(132,146)(133,142)(134,144)(135,143)(136,139)
(137,141)(138,140)(148,149)(151,167)(152,166)(153,168)(154,164)(155,163)
(156,165)(157,161)(158,160)(159,162)(169,171)(172,189)(173,188)(174,187)
(175,186)(176,185)(177,184)(178,183)(179,182)(180,181);
s2 := Sym(189)!(  1,  4)(  2,  6)(  3,  5)(  7, 19)(  8, 21)(  9, 20)( 10, 16)
( 11, 18)( 12, 17)( 14, 15)( 22, 67)( 23, 69)( 24, 68)( 25, 64)( 26, 66)
( 27, 65)( 28, 82)( 29, 84)( 30, 83)( 31, 79)( 32, 81)( 33, 80)( 34, 76)
( 35, 78)( 36, 77)( 37, 73)( 38, 75)( 39, 74)( 40, 70)( 41, 72)( 42, 71)
( 43,130)( 44,132)( 45,131)( 46,127)( 47,129)( 48,128)( 49,145)( 50,147)
( 51,146)( 52,142)( 53,144)( 54,143)( 55,139)( 56,141)( 57,140)( 58,136)
( 59,138)( 60,137)( 61,133)( 62,135)( 63,134)( 85, 89)( 86, 88)( 87, 90)
( 91,104)( 92,103)( 93,105)( 94,101)( 95,100)( 96,102)( 97, 98)(106,153)
(107,152)(108,151)(109,150)(110,149)(111,148)(112,168)(113,167)(114,166)
(115,165)(116,164)(117,163)(118,162)(119,161)(120,160)(121,159)(122,158)
(123,157)(124,156)(125,155)(126,154)(169,173)(170,172)(171,174)(175,188)
(176,187)(177,189)(178,185)(179,184)(180,186)(181,182);
poly := sub<Sym(189)|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*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2 >;

References : None.
to this polytope