Polytope of Type {84,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,6}*1512a
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 : 84
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Petrie
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}*216b
   21-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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, 68)( 23, 67)( 24, 69)( 25, 65)( 26, 64)( 27, 66)
( 28, 83)( 29, 82)( 30, 84)( 31, 80)( 32, 79)( 33, 81)( 34, 77)( 35, 76)
( 36, 78)( 37, 74)( 38, 73)( 39, 75)( 40, 71)( 41, 70)( 42, 72)( 43,131)
( 44,130)( 45,132)( 46,128)( 47,127)( 48,129)( 49,146)( 50,145)( 51,147)
( 52,143)( 53,142)( 54,144)( 55,140)( 56,139)( 57,141)( 58,137)( 59,136)
( 60,138)( 61,134)( 62,133)( 63,135)( 85, 90)( 86, 89)( 87, 88)( 91,105)
( 92,104)( 93,103)( 94,102)( 95,101)( 96,100)( 97, 99)(106,151)(107,153)
(108,152)(109,148)(110,150)(111,149)(112,166)(113,168)(114,167)(115,163)
(116,165)(117,164)(118,160)(119,162)(120,161)(121,157)(122,159)(123,158)
(124,154)(125,156)(126,155)(169,174)(170,173)(171,172)(175,189)(176,188)
(177,187)(178,186)(179,185)(180,184)(181,183);;
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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*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, 68)( 23, 67)( 24, 69)( 25, 65)( 26, 64)
( 27, 66)( 28, 83)( 29, 82)( 30, 84)( 31, 80)( 32, 79)( 33, 81)( 34, 77)
( 35, 76)( 36, 78)( 37, 74)( 38, 73)( 39, 75)( 40, 71)( 41, 70)( 42, 72)
( 43,131)( 44,130)( 45,132)( 46,128)( 47,127)( 48,129)( 49,146)( 50,145)
( 51,147)( 52,143)( 53,142)( 54,144)( 55,140)( 56,139)( 57,141)( 58,137)
( 59,136)( 60,138)( 61,134)( 62,133)( 63,135)( 85, 90)( 86, 89)( 87, 88)
( 91,105)( 92,104)( 93,103)( 94,102)( 95,101)( 96,100)( 97, 99)(106,151)
(107,153)(108,152)(109,148)(110,150)(111,149)(112,166)(113,168)(114,167)
(115,163)(116,165)(117,164)(118,160)(119,162)(120,161)(121,157)(122,159)
(123,158)(124,154)(125,156)(126,155)(169,174)(170,173)(171,172)(175,189)
(176,188)(177,187)(178,186)(179,185)(180,184)(181,183);
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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2 >;

References : None.
to this polytope