Questions?
See the FAQ
or other info.

Polytope of Type {6,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,84}*1512b
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 : 28
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}*216b
   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,  5)(  2,  4)(  3,  6)(  7, 20)(  8, 19)(  9, 21)( 10, 17)( 11, 16)
( 12, 18)( 13, 14)( 22,131)( 23,130)( 24,132)( 25,128)( 26,127)( 27,129)
( 28,146)( 29,145)( 30,147)( 31,143)( 32,142)( 33,144)( 34,140)( 35,139)
( 36,141)( 37,137)( 38,136)( 39,138)( 40,134)( 41,133)( 42,135)( 43, 68)
( 44, 67)( 45, 69)( 46, 65)( 47, 64)( 48, 66)( 49, 83)( 50, 82)( 51, 84)
( 52, 80)( 53, 79)( 54, 81)( 55, 77)( 56, 76)( 57, 78)( 58, 74)( 59, 73)
( 60, 75)( 61, 71)( 62, 70)( 63, 72)( 85,174)( 86,173)( 87,172)( 88,171)
( 89,170)( 90,169)( 91,189)( 92,188)( 93,187)( 94,186)( 95,185)( 96,184)
( 97,183)( 98,182)( 99,181)(100,180)(101,179)(102,178)(103,177)(104,176)
(105,175)(106,109)(107,111)(108,110)(112,124)(113,126)(114,125)(115,121)
(116,123)(117,122)(119,120)(148,151)(149,153)(150,152)(154,166)(155,168)
(156,167)(157,163)(158,165)(159,164)(161,162);;
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*s0*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0 ];;
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,  5)(  2,  4)(  3,  6)(  7, 20)(  8, 19)(  9, 21)( 10, 17)
( 11, 16)( 12, 18)( 13, 14)( 22,131)( 23,130)( 24,132)( 25,128)( 26,127)
( 27,129)( 28,146)( 29,145)( 30,147)( 31,143)( 32,142)( 33,144)( 34,140)
( 35,139)( 36,141)( 37,137)( 38,136)( 39,138)( 40,134)( 41,133)( 42,135)
( 43, 68)( 44, 67)( 45, 69)( 46, 65)( 47, 64)( 48, 66)( 49, 83)( 50, 82)
( 51, 84)( 52, 80)( 53, 79)( 54, 81)( 55, 77)( 56, 76)( 57, 78)( 58, 74)
( 59, 73)( 60, 75)( 61, 71)( 62, 70)( 63, 72)( 85,174)( 86,173)( 87,172)
( 88,171)( 89,170)( 90,169)( 91,189)( 92,188)( 93,187)( 94,186)( 95,185)
( 96,184)( 97,183)( 98,182)( 99,181)(100,180)(101,179)(102,178)(103,177)
(104,176)(105,175)(106,109)(107,111)(108,110)(112,124)(113,126)(114,125)
(115,121)(116,123)(117,122)(119,120)(148,151)(149,153)(150,152)(154,166)
(155,168)(156,167)(157,163)(158,165)(159,164)(161,162);
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*s0*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0 >; 
 
References : None.
to this polytope