Polytope of Type {6,138}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,138}*1656a
if this polytope has a name.
Group : SmallGroup(1656,116)
Rank : 3
Schlafli Type : {6,138}
Number of vertices, edges, etc : 6, 414, 138
Order of s₀s₁s₂ : 138
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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 : {6,46}*552
   9-fold quotients : {2,46}*184
   18-fold quotients : {2,23}*92
   23-fold quotients : {6,6}*72c
   46-fold quotients : {3,6}*36
   69-fold quotients : {6,2}*24
   138-fold quotients : {3,2}*12
   207-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 24, 47)( 25, 48)( 26, 49)( 27, 50)( 28, 51)( 29, 52)( 30, 53)( 31, 54)
( 32, 55)( 33, 56)( 34, 57)( 35, 58)( 36, 59)( 37, 60)( 38, 61)( 39, 62)
( 40, 63)( 41, 64)( 42, 65)( 43, 66)( 44, 67)( 45, 68)( 46, 69)( 70,139)
( 71,140)( 72,141)( 73,142)( 74,143)( 75,144)( 76,145)( 77,146)( 78,147)
( 79,148)( 80,149)( 81,150)( 82,151)( 83,152)( 84,153)( 85,154)( 86,155)
( 87,156)( 88,157)( 89,158)( 90,159)( 91,160)( 92,161)( 93,185)( 94,186)
( 95,187)( 96,188)( 97,189)( 98,190)( 99,191)(100,192)(101,193)(102,194)
(103,195)(104,196)(105,197)(106,198)(107,199)(108,200)(109,201)(110,202)
(111,203)(112,204)(113,205)(114,206)(115,207)(116,162)(117,163)(118,164)
(119,165)(120,166)(121,167)(122,168)(123,169)(124,170)(125,171)(126,172)
(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)(134,180)
(135,181)(136,182)(137,183)(138,184);;
s1 := (  1, 93)(  2,115)(  3,114)(  4,113)(  5,112)(  6,111)(  7,110)(  8,109)
(  9,108)( 10,107)( 11,106)( 12,105)( 13,104)( 14,103)( 15,102)( 16,101)
( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21, 96)( 22, 95)( 23, 94)( 24, 70)
( 25, 92)( 26, 91)( 27, 90)( 28, 89)( 29, 88)( 30, 87)( 31, 86)( 32, 85)
( 33, 84)( 34, 83)( 35, 82)( 36, 81)( 37, 80)( 38, 79)( 39, 78)( 40, 77)
( 41, 76)( 42, 75)( 43, 74)( 44, 73)( 45, 72)( 46, 71)( 47,116)( 48,138)
( 49,137)( 50,136)( 51,135)( 52,134)( 53,133)( 54,132)( 55,131)( 56,130)
( 57,129)( 58,128)( 59,127)( 60,126)( 61,125)( 62,124)( 63,123)( 64,122)
( 65,121)( 66,120)( 67,119)( 68,118)( 69,117)(139,162)(140,184)(141,183)
(142,182)(143,181)(144,180)(145,179)(146,178)(147,177)(148,176)(149,175)
(150,174)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)(157,167)
(158,166)(159,165)(160,164)(161,163)(186,207)(187,206)(188,205)(189,204)
(190,203)(191,202)(192,201)(193,200)(194,199)(195,198)(196,197);;
s2 := (  1,  2)(  3, 23)(  4, 22)(  5, 21)(  6, 20)(  7, 19)(  8, 18)(  9, 17)
( 10, 16)( 11, 15)( 12, 14)( 24, 25)( 26, 46)( 27, 45)( 28, 44)( 29, 43)
( 30, 42)( 31, 41)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 47, 48)( 49, 69)
( 50, 68)( 51, 67)( 52, 66)( 53, 65)( 54, 64)( 55, 63)( 56, 62)( 57, 61)
( 58, 60)( 70,140)( 71,139)( 72,161)( 73,160)( 74,159)( 75,158)( 76,157)
( 77,156)( 78,155)( 79,154)( 80,153)( 81,152)( 82,151)( 83,150)( 84,149)
( 85,148)( 86,147)( 87,146)( 88,145)( 89,144)( 90,143)( 91,142)( 92,141)
( 93,163)( 94,162)( 95,184)( 96,183)( 97,182)( 98,181)( 99,180)(100,179)
(101,178)(102,177)(103,176)(104,175)(105,174)(106,173)(107,172)(108,171)
(109,170)(110,169)(111,168)(112,167)(113,166)(114,165)(115,164)(116,186)
(117,185)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)(124,201)
(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)(132,193)
(133,192)(134,191)(135,190)(136,189)(137,188)(138,187);;
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*s1*s0*s1*s2*s0*s1*s2*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(207)!( 24, 47)( 25, 48)( 26, 49)( 27, 50)( 28, 51)( 29, 52)( 30, 53)
( 31, 54)( 32, 55)( 33, 56)( 34, 57)( 35, 58)( 36, 59)( 37, 60)( 38, 61)
( 39, 62)( 40, 63)( 41, 64)( 42, 65)( 43, 66)( 44, 67)( 45, 68)( 46, 69)
( 70,139)( 71,140)( 72,141)( 73,142)( 74,143)( 75,144)( 76,145)( 77,146)
( 78,147)( 79,148)( 80,149)( 81,150)( 82,151)( 83,152)( 84,153)( 85,154)
( 86,155)( 87,156)( 88,157)( 89,158)( 90,159)( 91,160)( 92,161)( 93,185)
( 94,186)( 95,187)( 96,188)( 97,189)( 98,190)( 99,191)(100,192)(101,193)
(102,194)(103,195)(104,196)(105,197)(106,198)(107,199)(108,200)(109,201)
(110,202)(111,203)(112,204)(113,205)(114,206)(115,207)(116,162)(117,163)
(118,164)(119,165)(120,166)(121,167)(122,168)(123,169)(124,170)(125,171)
(126,172)(127,173)(128,174)(129,175)(130,176)(131,177)(132,178)(133,179)
(134,180)(135,181)(136,182)(137,183)(138,184);
s1 := Sym(207)!(  1, 93)(  2,115)(  3,114)(  4,113)(  5,112)(  6,111)(  7,110)
(  8,109)(  9,108)( 10,107)( 11,106)( 12,105)( 13,104)( 14,103)( 15,102)
( 16,101)( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21, 96)( 22, 95)( 23, 94)
( 24, 70)( 25, 92)( 26, 91)( 27, 90)( 28, 89)( 29, 88)( 30, 87)( 31, 86)
( 32, 85)( 33, 84)( 34, 83)( 35, 82)( 36, 81)( 37, 80)( 38, 79)( 39, 78)
( 40, 77)( 41, 76)( 42, 75)( 43, 74)( 44, 73)( 45, 72)( 46, 71)( 47,116)
( 48,138)( 49,137)( 50,136)( 51,135)( 52,134)( 53,133)( 54,132)( 55,131)
( 56,130)( 57,129)( 58,128)( 59,127)( 60,126)( 61,125)( 62,124)( 63,123)
( 64,122)( 65,121)( 66,120)( 67,119)( 68,118)( 69,117)(139,162)(140,184)
(141,183)(142,182)(143,181)(144,180)(145,179)(146,178)(147,177)(148,176)
(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)
(157,167)(158,166)(159,165)(160,164)(161,163)(186,207)(187,206)(188,205)
(189,204)(190,203)(191,202)(192,201)(193,200)(194,199)(195,198)(196,197);
s2 := Sym(207)!(  1,  2)(  3, 23)(  4, 22)(  5, 21)(  6, 20)(  7, 19)(  8, 18)
(  9, 17)( 10, 16)( 11, 15)( 12, 14)( 24, 25)( 26, 46)( 27, 45)( 28, 44)
( 29, 43)( 30, 42)( 31, 41)( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 47, 48)
( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)( 54, 64)( 55, 63)( 56, 62)
( 57, 61)( 58, 60)( 70,140)( 71,139)( 72,161)( 73,160)( 74,159)( 75,158)
( 76,157)( 77,156)( 78,155)( 79,154)( 80,153)( 81,152)( 82,151)( 83,150)
( 84,149)( 85,148)( 86,147)( 87,146)( 88,145)( 89,144)( 90,143)( 91,142)
( 92,141)( 93,163)( 94,162)( 95,184)( 96,183)( 97,182)( 98,181)( 99,180)
(100,179)(101,178)(102,177)(103,176)(104,175)(105,174)(106,173)(107,172)
(108,171)(109,170)(110,169)(111,168)(112,167)(113,166)(114,165)(115,164)
(116,186)(117,185)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)
(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)
(132,193)(133,192)(134,191)(135,190)(136,189)(137,188)(138,187);
poly := sub<Sym(207)|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*s1*s0*s1*s2*s0*s1*s2*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 >;

References : None.
to this polytope