Polytope of Type {8,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,54}*864
Also Known As : {8,54|2}. if this polytope has another name.
Group : SmallGroup(864,126)
Rank : 3
Schlafli Type : {8,54}
Number of vertices, edges, etc : 8, 216, 54
Order of s₀s₁s₂ : 216
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,54,2} of size 1728
Vertex Figure Of :
   {2,8,54} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,54}*432a
   3-fold quotients : {8,18}*288
   4-fold quotients : {2,54}*216
   6-fold quotients : {4,18}*144a
   8-fold quotients : {2,27}*108
   9-fold quotients : {8,6}*96
   12-fold quotients : {2,18}*72
   18-fold quotients : {4,6}*48a
   24-fold quotients : {2,9}*36
   27-fold quotients : {8,2}*32
   36-fold quotients : {2,6}*24
   54-fold quotients : {4,2}*16
   72-fold quotients : {2,3}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,108}*1728a, {16,54}*1728
Permutation Representation (GAP) :

s0 := ( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)
( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)
( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)
( 79,106)( 80,107)( 81,108)(109,163)(110,164)(111,165)(112,166)(113,167)
(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)
(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)
(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)
(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)
(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)
(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)
(162,216);;
s1 := (  1,109)(  2,111)(  3,110)(  4,117)(  5,116)(  6,115)(  7,114)(  8,113)
(  9,112)( 10,135)( 11,134)( 12,133)( 13,132)( 14,131)( 15,130)( 16,129)
( 17,128)( 18,127)( 19,126)( 20,125)( 21,124)( 22,123)( 23,122)( 24,121)
( 25,120)( 26,119)( 27,118)( 28,136)( 29,138)( 30,137)( 31,144)( 32,143)
( 33,142)( 34,141)( 35,140)( 36,139)( 37,162)( 38,161)( 39,160)( 40,159)
( 41,158)( 42,157)( 43,156)( 44,155)( 45,154)( 46,153)( 47,152)( 48,151)
( 49,150)( 50,149)( 51,148)( 52,147)( 53,146)( 54,145)( 55,190)( 56,192)
( 57,191)( 58,198)( 59,197)( 60,196)( 61,195)( 62,194)( 63,193)( 64,216)
( 65,215)( 66,214)( 67,213)( 68,212)( 69,211)( 70,210)( 71,209)( 72,208)
( 73,207)( 74,206)( 75,205)( 76,204)( 77,203)( 78,202)( 79,201)( 80,200)
( 81,199)( 82,163)( 83,165)( 84,164)( 85,171)( 86,170)( 87,169)( 88,168)
( 89,167)( 90,166)( 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)(106,174)(107,173)(108,172);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 18)(  5, 17)(  6, 16)(  7, 15)(  8, 14)
(  9, 13)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 28, 37)( 29, 39)( 30, 38)
( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 46, 54)( 47, 53)
( 48, 52)( 49, 51)( 55, 64)( 56, 66)( 57, 65)( 58, 72)( 59, 71)( 60, 70)
( 61, 69)( 62, 68)( 63, 67)( 73, 81)( 74, 80)( 75, 79)( 76, 78)( 82, 91)
( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)( 90, 94)
(100,108)(101,107)(102,106)(103,105)(109,118)(110,120)(111,119)(112,126)
(113,125)(114,124)(115,123)(116,122)(117,121)(127,135)(128,134)(129,133)
(130,132)(136,145)(137,147)(138,146)(139,153)(140,152)(141,151)(142,150)
(143,149)(144,148)(154,162)(155,161)(156,160)(157,159)(163,172)(164,174)
(165,173)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(181,189)
(182,188)(183,187)(184,186)(190,199)(191,201)(192,200)(193,207)(194,206)
(195,205)(196,204)(197,203)(198,202)(208,216)(209,215)(210,214)(211,213);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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*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(216)!( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)
( 62, 89)( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)
( 70, 97)( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)
( 78,105)( 79,106)( 80,107)( 81,108)(109,163)(110,164)(111,165)(112,166)
(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)
(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)
(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)
(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)
(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)
(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)
(161,215)(162,216);
s1 := Sym(216)!(  1,109)(  2,111)(  3,110)(  4,117)(  5,116)(  6,115)(  7,114)
(  8,113)(  9,112)( 10,135)( 11,134)( 12,133)( 13,132)( 14,131)( 15,130)
( 16,129)( 17,128)( 18,127)( 19,126)( 20,125)( 21,124)( 22,123)( 23,122)
( 24,121)( 25,120)( 26,119)( 27,118)( 28,136)( 29,138)( 30,137)( 31,144)
( 32,143)( 33,142)( 34,141)( 35,140)( 36,139)( 37,162)( 38,161)( 39,160)
( 40,159)( 41,158)( 42,157)( 43,156)( 44,155)( 45,154)( 46,153)( 47,152)
( 48,151)( 49,150)( 50,149)( 51,148)( 52,147)( 53,146)( 54,145)( 55,190)
( 56,192)( 57,191)( 58,198)( 59,197)( 60,196)( 61,195)( 62,194)( 63,193)
( 64,216)( 65,215)( 66,214)( 67,213)( 68,212)( 69,211)( 70,210)( 71,209)
( 72,208)( 73,207)( 74,206)( 75,205)( 76,204)( 77,203)( 78,202)( 79,201)
( 80,200)( 81,199)( 82,163)( 83,165)( 84,164)( 85,171)( 86,170)( 87,169)
( 88,168)( 89,167)( 90,166)( 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)(106,174)(107,173)(108,172);
s2 := Sym(216)!(  1, 10)(  2, 12)(  3, 11)(  4, 18)(  5, 17)(  6, 16)(  7, 15)
(  8, 14)(  9, 13)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 28, 37)( 29, 39)
( 30, 38)( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 46, 54)
( 47, 53)( 48, 52)( 49, 51)( 55, 64)( 56, 66)( 57, 65)( 58, 72)( 59, 71)
( 60, 70)( 61, 69)( 62, 68)( 63, 67)( 73, 81)( 74, 80)( 75, 79)( 76, 78)
( 82, 91)( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)
( 90, 94)(100,108)(101,107)(102,106)(103,105)(109,118)(110,120)(111,119)
(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(127,135)(128,134)
(129,133)(130,132)(136,145)(137,147)(138,146)(139,153)(140,152)(141,151)
(142,150)(143,149)(144,148)(154,162)(155,161)(156,160)(157,159)(163,172)
(164,174)(165,173)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)
(181,189)(182,188)(183,187)(184,186)(190,199)(191,201)(192,200)(193,207)
(194,206)(195,205)(196,204)(197,203)(198,202)(208,216)(209,215)(210,214)
(211,213);
poly := sub<Sym(216)|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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope