Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728f
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}*864f
   3-fold quotients : {12,4}*576
   4-fold quotients : {6,12}*432f
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,12}*216b
   12-fold quotients : {6,4}*144
   24-fold quotients : {6,4}*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 := (  1,136)(  2,137)(  3,138)(  4,144)(  5,142)(  6,143)(  7,140)(  8,141)
(  9,139)( 10,156)( 11,154)( 12,155)( 13,161)( 14,162)( 15,160)( 16,157)
( 17,158)( 18,159)( 19,146)( 20,147)( 21,145)( 22,151)( 23,152)( 24,153)
( 25,150)( 26,148)( 27,149)( 28,109)( 29,110)( 30,111)( 31,117)( 32,115)
( 33,116)( 34,113)( 35,114)( 36,112)( 37,129)( 38,127)( 39,128)( 40,134)
( 41,135)( 42,133)( 43,130)( 44,131)( 45,132)( 46,119)( 47,120)( 48,118)
( 49,124)( 50,125)( 51,126)( 52,123)( 53,121)( 54,122)( 55,190)( 56,191)
( 57,192)( 58,198)( 59,196)( 60,197)( 61,194)( 62,195)( 63,193)( 64,210)
( 65,208)( 66,209)( 67,215)( 68,216)( 69,214)( 70,211)( 71,212)( 72,213)
( 73,200)( 74,201)( 75,199)( 76,205)( 77,206)( 78,207)( 79,204)( 80,202)
( 81,203)( 82,163)( 83,164)( 84,165)( 85,171)( 86,169)( 87,170)( 88,167)
( 89,168)( 90,166)( 91,183)( 92,181)( 93,182)( 94,188)( 95,189)( 96,187)
( 97,184)( 98,185)( 99,186)(100,173)(101,174)(102,172)(103,178)(104,179)
(105,180)(106,177)(107,175)(108,176);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 18)( 11, 17)( 12, 16)( 13, 15)
( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 48)( 49, 54)( 50, 53)( 51, 52)
( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 72)( 65, 71)( 66, 70)( 67, 69)
( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82, 85)( 83, 87)( 84, 86)( 89, 90)
( 91, 99)( 92, 98)( 93, 97)( 94, 96)(100,102)(103,108)(104,107)(105,106)
(109,193)(110,195)(111,194)(112,190)(113,192)(114,191)(115,196)(116,198)
(117,197)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)(124,201)
(125,200)(126,199)(127,210)(128,209)(129,208)(130,216)(131,215)(132,214)
(133,213)(134,212)(135,211)(136,166)(137,168)(138,167)(139,163)(140,165)
(141,164)(142,169)(143,171)(144,170)(145,180)(146,179)(147,178)(148,177)
(149,176)(150,175)(151,174)(152,173)(153,172)(154,183)(155,182)(156,181)
(157,189)(158,188)(159,187)(160,186)(161,185)(162,184);;
s2 := (  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);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*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*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope