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Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*864a
if this polytope has a name.
Group : SmallGroup(864,4000)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 36, 216, 72
Order of s0s1s2 : 6
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,12,2} of size 1728
Vertex Figure Of :
   {2,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,12}*432
   3-fold quotients : {6,12}*288b
   4-fold quotients : {6,6}*216c
   6-fold quotients : {3,12}*144
   8-fold quotients : {3,6}*108
   9-fold quotients : {6,4}*96
   12-fold quotients : {6,6}*72c
   18-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   24-fold quotients : {3,6}*36
   36-fold quotients : {3,4}*24, {6,2}*24
   72-fold quotients : {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12}*1728i, {12,12}*1728m, {6,24}*1728b, {6,24}*1728d, {6,12}*1728f
Permutation Representation (GAP) :
s0 := (  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)
( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 37, 73)
( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)( 43, 78)( 44, 80)( 45, 81)
( 46, 83)( 47, 82)( 48, 84)( 49, 97)( 50, 99)( 51, 98)( 52,100)( 53,101)
( 54,103)( 55,102)( 56,104)( 57,105)( 58,107)( 59,106)( 60,108)( 61, 85)
( 62, 87)( 63, 86)( 64, 88)( 65, 89)( 66, 91)( 67, 90)( 68, 92)( 69, 93)
( 70, 95)( 71, 94)( 72, 96)(110,111)(114,115)(118,119)(121,133)(122,135)
(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)(129,141)(130,143)
(131,142)(132,144)(145,181)(146,183)(147,182)(148,184)(149,185)(150,187)
(151,186)(152,188)(153,189)(154,191)(155,190)(156,192)(157,205)(158,207)
(159,206)(160,208)(161,209)(162,211)(163,210)(164,212)(165,213)(166,215)
(167,214)(168,216)(169,193)(170,195)(171,194)(172,196)(173,197)(174,199)
(175,198)(176,200)(177,201)(178,203)(179,202)(180,204);;
s1 := (  1,197)(  2,200)(  3,199)(  4,198)(  5,201)(  6,204)(  7,203)(  8,202)
(  9,193)( 10,196)( 11,195)( 12,194)( 13,189)( 14,192)( 15,191)( 16,190)
( 17,181)( 18,184)( 19,183)( 20,182)( 21,185)( 22,188)( 23,187)( 24,186)
( 25,205)( 26,208)( 27,207)( 28,206)( 29,209)( 30,212)( 31,211)( 32,210)
( 33,213)( 34,216)( 35,215)( 36,214)( 37,161)( 38,164)( 39,163)( 40,162)
( 41,165)( 42,168)( 43,167)( 44,166)( 45,157)( 46,160)( 47,159)( 48,158)
( 49,153)( 50,156)( 51,155)( 52,154)( 53,145)( 54,148)( 55,147)( 56,146)
( 57,149)( 58,152)( 59,151)( 60,150)( 61,169)( 62,172)( 63,171)( 64,170)
( 65,173)( 66,176)( 67,175)( 68,174)( 69,177)( 70,180)( 71,179)( 72,178)
( 73,125)( 74,128)( 75,127)( 76,126)( 77,129)( 78,132)( 79,131)( 80,130)
( 81,121)( 82,124)( 83,123)( 84,122)( 85,117)( 86,120)( 87,119)( 88,118)
( 89,109)( 90,112)( 91,111)( 92,110)( 93,113)( 94,116)( 95,115)( 96,114)
( 97,133)( 98,136)( 99,135)(100,134)(101,137)(102,140)(103,139)(104,138)
(105,141)(106,144)(107,143)(108,142);;
s2 := (  1,  4)(  2,  3)(  5, 12)(  6, 11)(  7, 10)(  8,  9)( 13, 28)( 14, 27)
( 15, 26)( 16, 25)( 17, 36)( 18, 35)( 19, 34)( 20, 33)( 21, 32)( 22, 31)
( 23, 30)( 24, 29)( 37, 40)( 38, 39)( 41, 48)( 42, 47)( 43, 46)( 44, 45)
( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 72)( 54, 71)( 55, 70)( 56, 69)
( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 73, 76)( 74, 75)( 77, 84)( 78, 83)
( 79, 82)( 80, 81)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89,108)( 90,107)
( 91,106)( 92,105)( 93,104)( 94,103)( 95,102)( 96,101)(109,112)(110,111)
(113,120)(114,119)(115,118)(116,117)(121,136)(122,135)(123,134)(124,133)
(125,144)(126,143)(127,142)(128,141)(129,140)(130,139)(131,138)(132,137)
(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(157,172)(158,171)
(159,170)(160,169)(161,180)(162,179)(163,178)(164,177)(165,176)(166,175)
(167,174)(168,173)(181,184)(182,183)(185,192)(186,191)(187,190)(188,189)
(193,208)(194,207)(195,206)(196,205)(197,216)(198,215)(199,214)(200,213)
(201,212)(202,211)(203,210)(204,209);;
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*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(216)!(  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)
( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)
( 37, 73)( 38, 75)( 39, 74)( 40, 76)( 41, 77)( 42, 79)( 43, 78)( 44, 80)
( 45, 81)( 46, 83)( 47, 82)( 48, 84)( 49, 97)( 50, 99)( 51, 98)( 52,100)
( 53,101)( 54,103)( 55,102)( 56,104)( 57,105)( 58,107)( 59,106)( 60,108)
( 61, 85)( 62, 87)( 63, 86)( 64, 88)( 65, 89)( 66, 91)( 67, 90)( 68, 92)
( 69, 93)( 70, 95)( 71, 94)( 72, 96)(110,111)(114,115)(118,119)(121,133)
(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)(129,141)
(130,143)(131,142)(132,144)(145,181)(146,183)(147,182)(148,184)(149,185)
(150,187)(151,186)(152,188)(153,189)(154,191)(155,190)(156,192)(157,205)
(158,207)(159,206)(160,208)(161,209)(162,211)(163,210)(164,212)(165,213)
(166,215)(167,214)(168,216)(169,193)(170,195)(171,194)(172,196)(173,197)
(174,199)(175,198)(176,200)(177,201)(178,203)(179,202)(180,204);
s1 := Sym(216)!(  1,197)(  2,200)(  3,199)(  4,198)(  5,201)(  6,204)(  7,203)
(  8,202)(  9,193)( 10,196)( 11,195)( 12,194)( 13,189)( 14,192)( 15,191)
( 16,190)( 17,181)( 18,184)( 19,183)( 20,182)( 21,185)( 22,188)( 23,187)
( 24,186)( 25,205)( 26,208)( 27,207)( 28,206)( 29,209)( 30,212)( 31,211)
( 32,210)( 33,213)( 34,216)( 35,215)( 36,214)( 37,161)( 38,164)( 39,163)
( 40,162)( 41,165)( 42,168)( 43,167)( 44,166)( 45,157)( 46,160)( 47,159)
( 48,158)( 49,153)( 50,156)( 51,155)( 52,154)( 53,145)( 54,148)( 55,147)
( 56,146)( 57,149)( 58,152)( 59,151)( 60,150)( 61,169)( 62,172)( 63,171)
( 64,170)( 65,173)( 66,176)( 67,175)( 68,174)( 69,177)( 70,180)( 71,179)
( 72,178)( 73,125)( 74,128)( 75,127)( 76,126)( 77,129)( 78,132)( 79,131)
( 80,130)( 81,121)( 82,124)( 83,123)( 84,122)( 85,117)( 86,120)( 87,119)
( 88,118)( 89,109)( 90,112)( 91,111)( 92,110)( 93,113)( 94,116)( 95,115)
( 96,114)( 97,133)( 98,136)( 99,135)(100,134)(101,137)(102,140)(103,139)
(104,138)(105,141)(106,144)(107,143)(108,142);
s2 := Sym(216)!(  1,  4)(  2,  3)(  5, 12)(  6, 11)(  7, 10)(  8,  9)( 13, 28)
( 14, 27)( 15, 26)( 16, 25)( 17, 36)( 18, 35)( 19, 34)( 20, 33)( 21, 32)
( 22, 31)( 23, 30)( 24, 29)( 37, 40)( 38, 39)( 41, 48)( 42, 47)( 43, 46)
( 44, 45)( 49, 64)( 50, 63)( 51, 62)( 52, 61)( 53, 72)( 54, 71)( 55, 70)
( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 73, 76)( 74, 75)( 77, 84)
( 78, 83)( 79, 82)( 80, 81)( 85,100)( 86, 99)( 87, 98)( 88, 97)( 89,108)
( 90,107)( 91,106)( 92,105)( 93,104)( 94,103)( 95,102)( 96,101)(109,112)
(110,111)(113,120)(114,119)(115,118)(116,117)(121,136)(122,135)(123,134)
(124,133)(125,144)(126,143)(127,142)(128,141)(129,140)(130,139)(131,138)
(132,137)(145,148)(146,147)(149,156)(150,155)(151,154)(152,153)(157,172)
(158,171)(159,170)(160,169)(161,180)(162,179)(163,178)(164,177)(165,176)
(166,175)(167,174)(168,173)(181,184)(182,183)(185,192)(186,191)(187,190)
(188,189)(193,208)(194,207)(195,206)(196,205)(197,216)(198,215)(199,214)
(200,213)(201,212)(202,211)(203,210)(204,209);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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 >; 
 
References : None.
to this polytope