Polytope of Type {6,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,84}*1512c
if this polytope has a name.
Group : SmallGroup(1512,827)
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}*216c
   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, 70)(  3, 69)(  4, 68)(  5, 67)(  6, 66)(  7, 65)(  8, 78)
(  9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 80)( 14, 79)( 15, 71)( 16, 77)
( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 85)( 23, 91)( 24, 90)
( 25, 89)( 26, 88)( 27, 87)( 28, 86)( 29, 99)( 30,105)( 31,104)( 32,103)
( 33,102)( 34,101)( 35,100)( 36, 92)( 37, 98)( 38, 97)( 39, 96)( 40, 95)
( 41, 94)( 42, 93)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)( 48,108)
( 49,107)( 50,120)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)( 56,121)
( 57,113)( 58,119)( 59,118)( 60,117)( 61,116)( 62,115)( 63,114)(128,133)
(129,132)(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)
(140,142)(149,154)(150,153)(151,152)(155,162)(156,168)(157,167)(158,166)
(159,165)(160,164)(161,163)(170,175)(171,174)(172,173)(176,183)(177,189)
(178,188)(179,187)(180,186)(181,185)(182,184);;
s2 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 16)
( 17, 21)( 18, 20)( 22, 72)( 23, 71)( 24, 77)( 25, 76)( 26, 75)( 27, 74)
( 28, 73)( 29, 65)( 30, 64)( 31, 70)( 32, 69)( 33, 68)( 34, 67)( 35, 66)
( 36, 79)( 37, 78)( 38, 84)( 39, 83)( 40, 82)( 41, 81)( 42, 80)( 43,135)
( 44,134)( 45,140)( 46,139)( 47,138)( 48,137)( 49,136)( 50,128)( 51,127)
( 52,133)( 53,132)( 54,131)( 55,130)( 56,129)( 57,142)( 58,141)( 59,147)
( 60,146)( 61,145)( 62,144)( 63,143)( 85, 93)( 86, 92)( 87, 98)( 88, 97)
( 89, 96)( 90, 95)( 91, 94)( 99,100)(101,105)(102,104)(106,156)(107,155)
(108,161)(109,160)(110,159)(111,158)(112,157)(113,149)(114,148)(115,154)
(116,153)(117,152)(118,151)(119,150)(120,163)(121,162)(122,168)(123,167)
(124,166)(125,165)(126,164)(169,177)(170,176)(171,182)(172,181)(173,180)
(174,179)(175,178)(183,184)(185,189)(186,188);;
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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, 70)(  3, 69)(  4, 68)(  5, 67)(  6, 66)(  7, 65)
(  8, 78)(  9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 80)( 14, 79)( 15, 71)
( 16, 77)( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 85)( 23, 91)
( 24, 90)( 25, 89)( 26, 88)( 27, 87)( 28, 86)( 29, 99)( 30,105)( 31,104)
( 32,103)( 33,102)( 34,101)( 35,100)( 36, 92)( 37, 98)( 38, 97)( 39, 96)
( 40, 95)( 41, 94)( 42, 93)( 43,106)( 44,112)( 45,111)( 46,110)( 47,109)
( 48,108)( 49,107)( 50,120)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)
( 56,121)( 57,113)( 58,119)( 59,118)( 60,117)( 61,116)( 62,115)( 63,114)
(128,133)(129,132)(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)
(139,143)(140,142)(149,154)(150,153)(151,152)(155,162)(156,168)(157,167)
(158,166)(159,165)(160,164)(161,163)(170,175)(171,174)(172,173)(176,183)
(177,189)(178,188)(179,187)(180,186)(181,185)(182,184);
s2 := Sym(189)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)
( 15, 16)( 17, 21)( 18, 20)( 22, 72)( 23, 71)( 24, 77)( 25, 76)( 26, 75)
( 27, 74)( 28, 73)( 29, 65)( 30, 64)( 31, 70)( 32, 69)( 33, 68)( 34, 67)
( 35, 66)( 36, 79)( 37, 78)( 38, 84)( 39, 83)( 40, 82)( 41, 81)( 42, 80)
( 43,135)( 44,134)( 45,140)( 46,139)( 47,138)( 48,137)( 49,136)( 50,128)
( 51,127)( 52,133)( 53,132)( 54,131)( 55,130)( 56,129)( 57,142)( 58,141)
( 59,147)( 60,146)( 61,145)( 62,144)( 63,143)( 85, 93)( 86, 92)( 87, 98)
( 88, 97)( 89, 96)( 90, 95)( 91, 94)( 99,100)(101,105)(102,104)(106,156)
(107,155)(108,161)(109,160)(110,159)(111,158)(112,157)(113,149)(114,148)
(115,154)(116,153)(117,152)(118,151)(119,150)(120,163)(121,162)(122,168)
(123,167)(124,166)(125,165)(126,164)(169,177)(170,176)(171,182)(172,181)
(173,180)(174,179)(175,178)(183,184)(185,189)(186,188);
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
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