Polytope of Type {6,27}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,27}*972a
if this polytope has a name.
Group : SmallGroup(972,107)
Rank : 3
Schlafli Type : {6,27}
Number of vertices, edges, etc : 18, 243, 81
Order of s0s1s2 : 54
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,27,2} of size 1944
Vertex Figure Of :
{2,6,27} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {6,9}*324a, {6,27}*324
9-fold quotients : {2,27}*108, {6,9}*108, {6,3}*108
27-fold quotients : {2,9}*36, {6,3}*36
81-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,54}*1944a
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)( 34, 58)
( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)( 42, 72)
( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)( 50, 80)
( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)(196,220)
(197,221)(198,222)(199,226)(200,227)(201,228)(202,232)(203,233)(204,234)
(205,229)(206,230)(207,231)(208,235)(209,236)(210,237)(211,241)(212,242)
(213,243)(214,238)(215,239)(216,240);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 31)( 5, 33)( 6, 32)( 7, 34)( 8, 36)
( 9, 35)( 10, 48)( 11, 47)( 12, 46)( 13, 51)( 14, 50)( 15, 49)( 16, 54)
( 17, 53)( 18, 52)( 19, 39)( 20, 38)( 21, 37)( 22, 42)( 23, 41)( 24, 40)
( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)
( 66, 73)( 67, 78)( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 82,210)
( 83,209)( 84,208)( 85,213)( 86,212)( 87,211)( 88,216)( 89,215)( 90,214)
( 91,201)( 92,200)( 93,199)( 94,204)( 95,203)( 96,202)( 97,207)( 98,206)
( 99,205)(100,192)(101,191)(102,190)(103,195)(104,194)(105,193)(106,198)
(107,197)(108,196)(109,183)(110,182)(111,181)(112,186)(113,185)(114,184)
(115,189)(116,188)(117,187)(118,174)(119,173)(120,172)(121,177)(122,176)
(123,175)(124,180)(125,179)(126,178)(127,165)(128,164)(129,163)(130,168)
(131,167)(132,166)(133,171)(134,170)(135,169)(136,237)(137,236)(138,235)
(139,240)(140,239)(141,238)(142,243)(143,242)(144,241)(145,228)(146,227)
(147,226)(148,231)(149,230)(150,229)(151,234)(152,233)(153,232)(154,219)
(155,218)(156,217)(157,222)(158,221)(159,220)(160,225)(161,224)(162,223);;
s2 := ( 1, 82)( 2, 84)( 3, 83)( 4, 85)( 5, 87)( 6, 86)( 7, 88)( 8, 90)
( 9, 89)( 10,102)( 11,101)( 12,100)( 13,105)( 14,104)( 15,103)( 16,108)
( 17,107)( 18,106)( 19, 93)( 20, 92)( 21, 91)( 22, 96)( 23, 95)( 24, 94)
( 25, 99)( 26, 98)( 27, 97)( 28,142)( 29,144)( 30,143)( 31,136)( 32,138)
( 33,137)( 34,139)( 35,141)( 36,140)( 37,162)( 38,161)( 39,160)( 40,156)
( 41,155)( 42,154)( 43,159)( 44,158)( 45,157)( 46,153)( 47,152)( 48,151)
( 49,147)( 50,146)( 51,145)( 52,150)( 53,149)( 54,148)( 55,112)( 56,114)
( 57,113)( 58,115)( 59,117)( 60,116)( 61,109)( 62,111)( 63,110)( 64,132)
( 65,131)( 66,130)( 67,135)( 68,134)( 69,133)( 70,129)( 71,128)( 72,127)
( 73,123)( 74,122)( 75,121)( 76,126)( 77,125)( 78,124)( 79,120)( 80,119)
( 81,118)(163,183)(164,182)(165,181)(166,186)(167,185)(168,184)(169,189)
(170,188)(171,187)(172,174)(175,177)(178,180)(190,243)(191,242)(192,241)
(193,237)(194,236)(195,235)(196,240)(197,239)(198,238)(199,234)(200,233)
(201,232)(202,228)(203,227)(204,226)(205,231)(206,230)(207,229)(208,225)
(209,224)(210,223)(211,219)(212,218)(213,217)(214,222)(215,221)(216,220);;
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,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(243)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)
( 34, 58)( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)
( 42, 72)( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)
( 50, 80)( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)
(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)
(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)
(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)
(135,159)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)
(196,220)(197,221)(198,222)(199,226)(200,227)(201,228)(202,232)(203,233)
(204,234)(205,229)(206,230)(207,231)(208,235)(209,236)(210,237)(211,241)
(212,242)(213,243)(214,238)(215,239)(216,240);
s1 := Sym(243)!( 1, 28)( 2, 30)( 3, 29)( 4, 31)( 5, 33)( 6, 32)( 7, 34)
( 8, 36)( 9, 35)( 10, 48)( 11, 47)( 12, 46)( 13, 51)( 14, 50)( 15, 49)
( 16, 54)( 17, 53)( 18, 52)( 19, 39)( 20, 38)( 21, 37)( 22, 42)( 23, 41)
( 24, 40)( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 59, 60)( 62, 63)( 64, 75)
( 65, 74)( 66, 73)( 67, 78)( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)
( 82,210)( 83,209)( 84,208)( 85,213)( 86,212)( 87,211)( 88,216)( 89,215)
( 90,214)( 91,201)( 92,200)( 93,199)( 94,204)( 95,203)( 96,202)( 97,207)
( 98,206)( 99,205)(100,192)(101,191)(102,190)(103,195)(104,194)(105,193)
(106,198)(107,197)(108,196)(109,183)(110,182)(111,181)(112,186)(113,185)
(114,184)(115,189)(116,188)(117,187)(118,174)(119,173)(120,172)(121,177)
(122,176)(123,175)(124,180)(125,179)(126,178)(127,165)(128,164)(129,163)
(130,168)(131,167)(132,166)(133,171)(134,170)(135,169)(136,237)(137,236)
(138,235)(139,240)(140,239)(141,238)(142,243)(143,242)(144,241)(145,228)
(146,227)(147,226)(148,231)(149,230)(150,229)(151,234)(152,233)(153,232)
(154,219)(155,218)(156,217)(157,222)(158,221)(159,220)(160,225)(161,224)
(162,223);
s2 := Sym(243)!( 1, 82)( 2, 84)( 3, 83)( 4, 85)( 5, 87)( 6, 86)( 7, 88)
( 8, 90)( 9, 89)( 10,102)( 11,101)( 12,100)( 13,105)( 14,104)( 15,103)
( 16,108)( 17,107)( 18,106)( 19, 93)( 20, 92)( 21, 91)( 22, 96)( 23, 95)
( 24, 94)( 25, 99)( 26, 98)( 27, 97)( 28,142)( 29,144)( 30,143)( 31,136)
( 32,138)( 33,137)( 34,139)( 35,141)( 36,140)( 37,162)( 38,161)( 39,160)
( 40,156)( 41,155)( 42,154)( 43,159)( 44,158)( 45,157)( 46,153)( 47,152)
( 48,151)( 49,147)( 50,146)( 51,145)( 52,150)( 53,149)( 54,148)( 55,112)
( 56,114)( 57,113)( 58,115)( 59,117)( 60,116)( 61,109)( 62,111)( 63,110)
( 64,132)( 65,131)( 66,130)( 67,135)( 68,134)( 69,133)( 70,129)( 71,128)
( 72,127)( 73,123)( 74,122)( 75,121)( 76,126)( 77,125)( 78,124)( 79,120)
( 80,119)( 81,118)(163,183)(164,182)(165,181)(166,186)(167,185)(168,184)
(169,189)(170,188)(171,187)(172,174)(175,177)(178,180)(190,243)(191,242)
(192,241)(193,237)(194,236)(195,235)(196,240)(197,239)(198,238)(199,234)
(200,233)(201,232)(202,228)(203,227)(204,226)(205,231)(206,230)(207,229)
(208,225)(209,224)(210,223)(211,219)(212,218)(213,217)(214,222)(215,221)
(216,220);
poly := sub<Sym(243)|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,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*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 >;
References : None.
to this polytope