Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728d
if this polytope has a name.
Group : SmallGroup(1728,12630)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,12}*864e
   3-fold quotients : {4,12}*576
   4-fold quotients : {12,6}*432f
   6-fold quotients : {4,12}*288
   8-fold quotients : {12,6}*216a
   12-fold quotients : {4,6}*144
   24-fold quotients : {4,6}*72
   27-fold quotients : {4,4}*64
   54-fold quotients : {4,4}*32
   108-fold quotients : {2,4}*16, {4,2}*16
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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