Polytope of Type {6,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*1944
if this polytope has a name.
Group : SmallGroup(1944,2323)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 27, 486, 162
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
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 : {6,36}*648a, {6,12}*648, {6,36}*648b, {6,36}*648c
9-fold quotients : {6,4}*216, {6,12}*216a, {6,12}*216b, {6,12}*216c
27-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)
( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)
( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)
( 52, 79)( 53, 80)( 54, 81)( 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)(109,217)(110,218)
(111,219)(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)
(119,227)(120,228)(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)
(127,235)(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)(134,242)
(135,243)(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, 82)( 2, 84)( 3, 83)( 4, 88)( 5, 90)( 6, 89)( 7, 85)( 8, 87)
( 9, 86)( 10,103)( 11,105)( 12,104)( 13,100)( 14,102)( 15,101)( 16,106)
( 17,108)( 18,107)( 19, 94)( 20, 96)( 21, 95)( 22, 91)( 23, 93)( 24, 92)
( 25, 97)( 26, 99)( 27, 98)( 28,110)( 29,109)( 30,111)( 31,116)( 32,115)
( 33,117)( 34,113)( 35,112)( 36,114)( 37,131)( 38,130)( 39,132)( 40,128)
( 41,127)( 42,129)( 43,134)( 44,133)( 45,135)( 46,122)( 47,121)( 48,123)
( 49,119)( 50,118)( 51,120)( 52,125)( 53,124)( 54,126)( 55,138)( 56,137)
( 57,136)( 58,144)( 59,143)( 60,142)( 61,141)( 62,140)( 63,139)( 64,159)
( 65,158)( 66,157)( 67,156)( 68,155)( 69,154)( 70,162)( 71,161)( 72,160)
( 73,150)( 74,149)( 75,148)( 76,147)( 77,146)( 78,145)( 79,153)( 80,152)
( 81,151)(164,165)(166,169)(167,171)(168,170)(172,184)(173,186)(174,185)
(175,181)(176,183)(177,182)(178,187)(179,189)(180,188)(190,191)(193,197)
(194,196)(195,198)(199,212)(200,211)(201,213)(202,209)(203,208)(204,210)
(205,215)(206,214)(207,216)(217,219)(220,225)(221,224)(222,223)(226,240)
(227,239)(228,238)(229,237)(230,236)(231,235)(232,243)(233,242)(234,241);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)( 8, 15)
( 9, 14)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 91)( 29, 93)( 30, 92)
( 31, 97)( 32, 99)( 33, 98)( 34, 94)( 35, 96)( 36, 95)( 37, 82)( 38, 84)
( 39, 83)( 40, 88)( 41, 90)( 42, 89)( 43, 85)( 44, 87)( 45, 86)( 46,103)
( 47,105)( 48,104)( 49,100)( 50,102)( 51,101)( 52,106)( 53,108)( 54,107)
( 55,172)( 56,174)( 57,173)( 58,178)( 59,180)( 60,179)( 61,175)( 62,177)
( 63,176)( 64,163)( 65,165)( 66,164)( 67,169)( 68,171)( 69,170)( 70,166)
( 71,168)( 72,167)( 73,184)( 74,186)( 75,185)( 76,181)( 77,183)( 78,182)
( 79,187)( 80,189)( 81,188)(109,119)(110,118)(111,120)(112,125)(113,124)
(114,126)(115,122)(116,121)(117,123)(127,131)(128,130)(129,132)(133,134)
(136,201)(137,200)(138,199)(139,207)(140,206)(141,205)(142,204)(143,203)
(144,202)(145,192)(146,191)(147,190)(148,198)(149,197)(150,196)(151,195)
(152,194)(153,193)(154,213)(155,212)(156,211)(157,210)(158,209)(159,208)
(160,216)(161,215)(162,214)(217,227)(218,226)(219,228)(220,233)(221,232)
(222,234)(223,230)(224,229)(225,231)(235,239)(236,238)(237,240)(241,242);;
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*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*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*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(243)!( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)
( 35, 62)( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)
( 43, 70)( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)
( 51, 78)( 52, 79)( 53, 80)( 54, 81)( 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)(109,217)
(110,218)(111,219)(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)
(118,226)(119,227)(120,228)(121,229)(122,230)(123,231)(124,232)(125,233)
(126,234)(127,235)(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)
(134,242)(135,243)(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(243)!( 1, 82)( 2, 84)( 3, 83)( 4, 88)( 5, 90)( 6, 89)( 7, 85)
( 8, 87)( 9, 86)( 10,103)( 11,105)( 12,104)( 13,100)( 14,102)( 15,101)
( 16,106)( 17,108)( 18,107)( 19, 94)( 20, 96)( 21, 95)( 22, 91)( 23, 93)
( 24, 92)( 25, 97)( 26, 99)( 27, 98)( 28,110)( 29,109)( 30,111)( 31,116)
( 32,115)( 33,117)( 34,113)( 35,112)( 36,114)( 37,131)( 38,130)( 39,132)
( 40,128)( 41,127)( 42,129)( 43,134)( 44,133)( 45,135)( 46,122)( 47,121)
( 48,123)( 49,119)( 50,118)( 51,120)( 52,125)( 53,124)( 54,126)( 55,138)
( 56,137)( 57,136)( 58,144)( 59,143)( 60,142)( 61,141)( 62,140)( 63,139)
( 64,159)( 65,158)( 66,157)( 67,156)( 68,155)( 69,154)( 70,162)( 71,161)
( 72,160)( 73,150)( 74,149)( 75,148)( 76,147)( 77,146)( 78,145)( 79,153)
( 80,152)( 81,151)(164,165)(166,169)(167,171)(168,170)(172,184)(173,186)
(174,185)(175,181)(176,183)(177,182)(178,187)(179,189)(180,188)(190,191)
(193,197)(194,196)(195,198)(199,212)(200,211)(201,213)(202,209)(203,208)
(204,210)(205,215)(206,214)(207,216)(217,219)(220,225)(221,224)(222,223)
(226,240)(227,239)(228,238)(229,237)(230,236)(231,235)(232,243)(233,242)
(234,241);
s2 := Sym(243)!( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)
( 8, 15)( 9, 14)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 91)( 29, 93)
( 30, 92)( 31, 97)( 32, 99)( 33, 98)( 34, 94)( 35, 96)( 36, 95)( 37, 82)
( 38, 84)( 39, 83)( 40, 88)( 41, 90)( 42, 89)( 43, 85)( 44, 87)( 45, 86)
( 46,103)( 47,105)( 48,104)( 49,100)( 50,102)( 51,101)( 52,106)( 53,108)
( 54,107)( 55,172)( 56,174)( 57,173)( 58,178)( 59,180)( 60,179)( 61,175)
( 62,177)( 63,176)( 64,163)( 65,165)( 66,164)( 67,169)( 68,171)( 69,170)
( 70,166)( 71,168)( 72,167)( 73,184)( 74,186)( 75,185)( 76,181)( 77,183)
( 78,182)( 79,187)( 80,189)( 81,188)(109,119)(110,118)(111,120)(112,125)
(113,124)(114,126)(115,122)(116,121)(117,123)(127,131)(128,130)(129,132)
(133,134)(136,201)(137,200)(138,199)(139,207)(140,206)(141,205)(142,204)
(143,203)(144,202)(145,192)(146,191)(147,190)(148,198)(149,197)(150,196)
(151,195)(152,194)(153,193)(154,213)(155,212)(156,211)(157,210)(158,209)
(159,208)(160,216)(161,215)(162,214)(217,227)(218,226)(219,228)(220,233)
(221,232)(222,234)(223,230)(224,229)(225,231)(235,239)(236,238)(237,240)
(241,242);
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*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*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*s2*s0*s1 >;
References : None.
to this polytope