Polytope of Type {84,6}

Play with this polytope as a twisty puzzle

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

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4, 19)(  5, 21)(  6, 20)(  7, 16)(  8, 18)(  9, 17)( 10, 13)( 11, 15)( 12, 14)( 23, 24)( 25, 40)( 26, 42)( 27, 41)( 28, 37)( 29, 39)( 30, 38)( 31, 34)( 32, 36)( 33, 35)( 44, 45)( 46, 61)( 47, 63)( 48, 62)( 49, 58)( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 64,127)( 65,129)( 66,128)( 67,145)( 68,147)( 69,146)( 70,142)( 71,144)( 72,143)( 73,139)( 74,141)( 75,140)( 76,136)( 77,138)( 78,137)( 79,133)( 80,135)( 81,134)( 82,130)( 83,132)( 84,131)( 85,148)( 86,150)( 87,149)( 88,166)( 89,168)( 90,167)( 91,163)( 92,165)( 93,164)( 94,160)( 95,162)( 96,161)( 97,157)( 98,159)( 99,158)(100,154)(101,156)(102,155)(103,151)(104,153)(105,152)(106,169)(107,171)(108,170)(109,187)(110,189)(111,188)(112,184)(113,186)(114,185)(115,181)(116,183)(117,182)(118,178)(119,180)(120,179)(121,175)(122,177)(123,176)(124,172)(125,174)(126,173);;
s1 := (  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);;
s2 := (  1, 22)(  2, 23)(  3, 24)(  4, 25)(  5, 26)(  6, 27)(  7, 28)(  8, 29)(  9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)( 15, 36)( 16, 37)( 17, 38)( 18, 39)( 19, 40)( 20, 41)( 21, 42)( 64,148)( 65,149)( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,155)( 72,156)( 73,157)( 74,158)( 75,159)( 76,160)( 77,161)( 78,162)( 79,163)( 80,164)( 81,165)( 82,166)( 83,167)( 84,168)( 85,127)( 86,128)( 87,129)( 88,130)( 89,131)( 90,132)( 91,133)( 92,134)( 93,135)( 94,136)( 95,137)( 96,138)( 97,139)( 98,140)( 99,141)(100,142)(101,143)(102,144)(103,145)(104,146)(105,147)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189);;
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*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(189)!(  2,  3)(  4, 19)(  5, 21)(  6, 20)(  7, 16)(  8, 18)(  9, 17)( 10, 13)( 11, 15)( 12, 14)( 23, 24)( 25, 40)( 26, 42)( 27, 41)( 28, 37)( 29, 39)( 30, 38)( 31, 34)( 32, 36)( 33, 35)( 44, 45)( 46, 61)( 47, 63)( 48, 62)( 49, 58)( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 64,127)( 65,129)( 66,128)( 67,145)( 68,147)( 69,146)( 70,142)( 71,144)( 72,143)( 73,139)( 74,141)( 75,140)( 76,136)( 77,138)( 78,137)( 79,133)( 80,135)( 81,134)( 82,130)( 83,132)( 84,131)( 85,148)( 86,150)( 87,149)( 88,166)( 89,168)( 90,167)( 91,163)( 92,165)( 93,164)( 94,160)( 95,162)( 96,161)( 97,157)( 98,159)( 99,158)(100,154)(101,156)(102,155)(103,151)(104,153)(105,152)(106,169)(107,171)(108,170)(109,187)(110,189)(111,188)(112,184)(113,186)(114,185)(115,181)(116,183)(117,182)(118,178)(119,180)(120,179)(121,175)(122,177)(123,176)(124,172)(125,174)(126,173);
s1 := 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);
s2 := Sym(189)!(  1, 22)(  2, 23)(  3, 24)(  4, 25)(  5, 26)(  6, 27)(  7, 28)(  8, 29)(  9, 30)( 10, 31)( 11, 32)( 12, 33)( 13, 34)( 14, 35)( 15, 36)( 16, 37)( 17, 38)( 18, 39)( 19, 40)( 20, 41)( 21, 42)( 64,148)( 65,149)( 66,150)( 67,151)( 68,152)( 69,153)( 70,154)( 71,155)( 72,156)( 73,157)( 74,158)( 75,159)( 76,160)( 77,161)( 78,162)( 79,163)( 80,164)( 81,165)( 82,166)( 83,167)( 84,168)( 85,127)( 86,128)( 87,129)( 88,130)( 89,131)( 90,132)( 91,133)( 92,134)( 93,135)( 94,136)( 95,137)( 96,138)( 97,139)( 98,140)( 99,141)(100,142)(101,143)(102,144)(103,145)(104,146)(105,147)(106,169)(107,170)(108,171)(109,172)(110,173)(111,174)(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2 >;
References : None.
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