Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728t
if this polytope has a name.
Group : SmallGroup(1728,33548)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   2-fold quotients : {12,12}*864l
   3-fold quotients : {12,4}*576
   4-fold quotients : {6,12}*432i
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,12}*216c
   9-fold quotients : {4,12}*192a
   12-fold quotients : {6,4}*144
   18-fold quotients : {4,12}*96a
   24-fold quotients : {6,4}*72
   27-fold quotients : {4,4}*64
   36-fold quotients : {2,12}*48, {4,6}*48a
   54-fold quotients : {4,4}*32
   72-fold quotients : {2,6}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {2,3}*12
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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