Polytope of Type {54,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,8}*864
Also Known As : {54,8|2}. if this polytope has another name.
Group : SmallGroup(864,126)
Rank : 3
Schlafli Type : {54,8}
Number of vertices, edges, etc : 54, 216, 8
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 :
{54,8,2} of size 1728
Vertex Figure Of :
{2,54,8} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {54,4}*432a
3-fold quotients : {18,8}*288
4-fold quotients : {54,2}*216
6-fold quotients : {18,4}*144a
8-fold quotients : {27,2}*108
9-fold quotients : {6,8}*96
12-fold quotients : {18,2}*72
18-fold quotients : {6,4}*48a
24-fold quotients : {9,2}*36
27-fold quotients : {2,8}*32
36-fold quotients : {6,2}*24
54-fold quotients : {2,4}*16
72-fold quotients : {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {108,8}*1728a, {54,16}*1728
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)( 32, 35)
( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)
( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)( 65, 80)
( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 83, 84)
( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)( 95,104)
( 96,103)( 97,102)( 98,101)( 99,100)(110,111)(112,117)(113,116)(114,115)
(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)
(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)(147,160)
(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(164,165)(166,171)
(167,170)(168,169)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)
(178,183)(179,182)(180,181)(191,192)(193,198)(194,197)(195,196)(199,216)
(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208);;
s1 := ( 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, 91)( 56, 93)( 57, 92)( 58, 99)( 59, 98)( 60, 97)
( 61, 96)( 62, 95)( 63, 94)( 64, 82)( 65, 84)( 66, 83)( 67, 90)( 68, 89)
( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73,108)( 74,107)( 75,106)( 76,105)
( 77,104)( 78,103)( 79,102)( 80,101)( 81,100)(109,172)(110,174)(111,173)
(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,163)(119,165)
(120,164)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)(127,189)
(128,188)(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)
(136,199)(137,201)(138,200)(139,207)(140,206)(141,205)(142,204)(143,203)
(144,202)(145,190)(146,192)(147,191)(148,198)(149,197)(150,196)(151,195)
(152,194)(153,193)(154,216)(155,215)(156,214)(157,213)(158,212)(159,211)
(160,210)(161,209)(162,208);;
s2 := ( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)
( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)
( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)
( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)
( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)
( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)
( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)
( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)
(105,186)(106,187)(107,188)(108,189);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)
( 32, 35)( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)
( 43, 48)( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)
( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(110,111)(112,117)(113,116)
(114,115)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)
(125,128)(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)
(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(164,165)
(166,171)(167,170)(168,169)(172,189)(173,188)(174,187)(175,186)(176,185)
(177,184)(178,183)(179,182)(180,181)(191,192)(193,198)(194,197)(195,196)
(199,216)(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)
(207,208);
s1 := 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, 91)( 56, 93)( 57, 92)( 58, 99)( 59, 98)
( 60, 97)( 61, 96)( 62, 95)( 63, 94)( 64, 82)( 65, 84)( 66, 83)( 67, 90)
( 68, 89)( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73,108)( 74,107)( 75,106)
( 76,105)( 77,104)( 78,103)( 79,102)( 80,101)( 81,100)(109,172)(110,174)
(111,173)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,163)
(119,165)(120,164)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)
(127,189)(128,188)(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)
(135,181)(136,199)(137,201)(138,200)(139,207)(140,206)(141,205)(142,204)
(143,203)(144,202)(145,190)(146,192)(147,191)(148,198)(149,197)(150,196)
(151,195)(152,194)(153,193)(154,216)(155,215)(156,214)(157,213)(158,212)
(159,211)(160,210)(161,209)(162,208);
s2 := Sym(216)!( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)
( 8,116)( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)
( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)
( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)
( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)
( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)
( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)
( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)
( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)
( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)
(104,185)(105,186)(106,187)(107,188)(108,189);
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope