Polytope of Type {2,12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,12}*1728j
if this polytope has a name.
Group : SmallGroup(1728,46611)
Rank : 4
Schlafli Type : {2,12,12}
Number of vertices, edges, etc : 2, 36, 216, 36
Order of s₀s₁s₂s₃ : 4
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
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 : {2,6,12}*864h
   3-fold quotients : {2,12,4}*576
   6-fold quotients : {2,6,4}*288
   12-fold quotients : {2,6,4}*144
   27-fold quotients : {2,4,4}*64
   54-fold quotients : {2,2,4}*32, {2,4,2}*32
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  3, 57)(  4, 59)(  5, 58)(  6, 63)(  7, 65)(  8, 64)(  9, 60)( 10, 62)
( 11, 61)( 12, 75)( 13, 77)( 14, 76)( 15, 81)( 16, 83)( 17, 82)( 18, 78)
( 19, 80)( 20, 79)( 21, 66)( 22, 68)( 23, 67)( 24, 72)( 25, 74)( 26, 73)
( 27, 69)( 28, 71)( 29, 70)( 30, 84)( 31, 86)( 32, 85)( 33, 90)( 34, 92)
( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39,102)( 40,104)( 41,103)( 42,108)
( 43,110)( 44,109)( 45,105)( 46,107)( 47,106)( 48, 93)( 49, 95)( 50, 94)
( 51, 99)( 52,101)( 53,100)( 54, 96)( 55, 98)( 56, 97)(111,192)(112,194)
(113,193)(114,198)(115,200)(116,199)(117,195)(118,197)(119,196)(120,210)
(121,212)(122,211)(123,216)(124,218)(125,217)(126,213)(127,215)(128,214)
(129,201)(130,203)(131,202)(132,207)(133,209)(134,208)(135,204)(136,206)
(137,205)(138,165)(139,167)(140,166)(141,171)(142,173)(143,172)(144,168)
(145,170)(146,169)(147,183)(148,185)(149,184)(150,189)(151,191)(152,190)
(153,186)(154,188)(155,187)(156,174)(157,176)(158,175)(159,180)(160,182)
(161,181)(162,177)(163,179)(164,178);;
s2 := (  3,169)(  4,168)(  5,170)(  6,166)(  7,165)(  8,167)(  9,172)( 10,171)
( 11,173)( 12,181)( 13,180)( 14,182)( 15,178)( 16,177)( 17,179)( 18,175)
( 19,174)( 20,176)( 21,184)( 22,183)( 23,185)( 24,190)( 25,189)( 26,191)
( 27,187)( 28,186)( 29,188)( 30,196)( 31,195)( 32,197)( 33,193)( 34,192)
( 35,194)( 36,199)( 37,198)( 38,200)( 39,208)( 40,207)( 41,209)( 42,205)
( 43,204)( 44,206)( 45,202)( 46,201)( 47,203)( 48,211)( 49,210)( 50,212)
( 51,217)( 52,216)( 53,218)( 54,214)( 55,213)( 56,215)( 57,115)( 58,114)
( 59,116)( 60,112)( 61,111)( 62,113)( 63,118)( 64,117)( 65,119)( 66,127)
( 67,126)( 68,128)( 69,124)( 70,123)( 71,125)( 72,121)( 73,120)( 74,122)
( 75,130)( 76,129)( 77,131)( 78,136)( 79,135)( 80,137)( 81,133)( 82,132)
( 83,134)( 84,142)( 85,141)( 86,143)( 87,139)( 88,138)( 89,140)( 90,145)
( 91,144)( 92,146)( 93,154)( 94,153)( 95,155)( 96,151)( 97,150)( 98,152)
( 99,148)(100,147)(101,149)(102,157)(103,156)(104,158)(105,163)(106,162)
(107,164)(108,160)(109,159)(110,161);;
s3 := (  4,  5)(  6, 24)(  7, 26)(  8, 25)(  9, 18)( 10, 20)( 11, 19)( 12, 21)
( 13, 23)( 14, 22)( 16, 17)( 28, 29)( 31, 32)( 33, 51)( 34, 53)( 35, 52)
( 36, 45)( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 43, 44)( 55, 56)
( 58, 59)( 60, 78)( 61, 80)( 62, 79)( 63, 72)( 64, 74)( 65, 73)( 66, 75)
( 67, 77)( 68, 76)( 70, 71)( 82, 83)( 85, 86)( 87,105)( 88,107)( 89,106)
( 90, 99)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 97, 98)(109,110)
(111,192)(112,194)(113,193)(114,213)(115,215)(116,214)(117,207)(118,209)
(119,208)(120,210)(121,212)(122,211)(123,204)(124,206)(125,205)(126,198)
(127,200)(128,199)(129,201)(130,203)(131,202)(132,195)(133,197)(134,196)
(135,216)(136,218)(137,217)(138,165)(139,167)(140,166)(141,186)(142,188)
(143,187)(144,180)(145,182)(146,181)(147,183)(148,185)(149,184)(150,177)
(151,179)(152,178)(153,171)(154,173)(155,172)(156,174)(157,176)(158,175)
(159,168)(160,170)(161,169)(162,189)(163,191)(164,190);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  3, 57)(  4, 59)(  5, 58)(  6, 63)(  7, 65)(  8, 64)(  9, 60)
( 10, 62)( 11, 61)( 12, 75)( 13, 77)( 14, 76)( 15, 81)( 16, 83)( 17, 82)
( 18, 78)( 19, 80)( 20, 79)( 21, 66)( 22, 68)( 23, 67)( 24, 72)( 25, 74)
( 26, 73)( 27, 69)( 28, 71)( 29, 70)( 30, 84)( 31, 86)( 32, 85)( 33, 90)
( 34, 92)( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39,102)( 40,104)( 41,103)
( 42,108)( 43,110)( 44,109)( 45,105)( 46,107)( 47,106)( 48, 93)( 49, 95)
( 50, 94)( 51, 99)( 52,101)( 53,100)( 54, 96)( 55, 98)( 56, 97)(111,192)
(112,194)(113,193)(114,198)(115,200)(116,199)(117,195)(118,197)(119,196)
(120,210)(121,212)(122,211)(123,216)(124,218)(125,217)(126,213)(127,215)
(128,214)(129,201)(130,203)(131,202)(132,207)(133,209)(134,208)(135,204)
(136,206)(137,205)(138,165)(139,167)(140,166)(141,171)(142,173)(143,172)
(144,168)(145,170)(146,169)(147,183)(148,185)(149,184)(150,189)(151,191)
(152,190)(153,186)(154,188)(155,187)(156,174)(157,176)(158,175)(159,180)
(160,182)(161,181)(162,177)(163,179)(164,178);
s2 := Sym(218)!(  3,169)(  4,168)(  5,170)(  6,166)(  7,165)(  8,167)(  9,172)
( 10,171)( 11,173)( 12,181)( 13,180)( 14,182)( 15,178)( 16,177)( 17,179)
( 18,175)( 19,174)( 20,176)( 21,184)( 22,183)( 23,185)( 24,190)( 25,189)
( 26,191)( 27,187)( 28,186)( 29,188)( 30,196)( 31,195)( 32,197)( 33,193)
( 34,192)( 35,194)( 36,199)( 37,198)( 38,200)( 39,208)( 40,207)( 41,209)
( 42,205)( 43,204)( 44,206)( 45,202)( 46,201)( 47,203)( 48,211)( 49,210)
( 50,212)( 51,217)( 52,216)( 53,218)( 54,214)( 55,213)( 56,215)( 57,115)
( 58,114)( 59,116)( 60,112)( 61,111)( 62,113)( 63,118)( 64,117)( 65,119)
( 66,127)( 67,126)( 68,128)( 69,124)( 70,123)( 71,125)( 72,121)( 73,120)
( 74,122)( 75,130)( 76,129)( 77,131)( 78,136)( 79,135)( 80,137)( 81,133)
( 82,132)( 83,134)( 84,142)( 85,141)( 86,143)( 87,139)( 88,138)( 89,140)
( 90,145)( 91,144)( 92,146)( 93,154)( 94,153)( 95,155)( 96,151)( 97,150)
( 98,152)( 99,148)(100,147)(101,149)(102,157)(103,156)(104,158)(105,163)
(106,162)(107,164)(108,160)(109,159)(110,161);
s3 := Sym(218)!(  4,  5)(  6, 24)(  7, 26)(  8, 25)(  9, 18)( 10, 20)( 11, 19)
( 12, 21)( 13, 23)( 14, 22)( 16, 17)( 28, 29)( 31, 32)( 33, 51)( 34, 53)
( 35, 52)( 36, 45)( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 43, 44)
( 55, 56)( 58, 59)( 60, 78)( 61, 80)( 62, 79)( 63, 72)( 64, 74)( 65, 73)
( 66, 75)( 67, 77)( 68, 76)( 70, 71)( 82, 83)( 85, 86)( 87,105)( 88,107)
( 89,106)( 90, 99)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 97, 98)
(109,110)(111,192)(112,194)(113,193)(114,213)(115,215)(116,214)(117,207)
(118,209)(119,208)(120,210)(121,212)(122,211)(123,204)(124,206)(125,205)
(126,198)(127,200)(128,199)(129,201)(130,203)(131,202)(132,195)(133,197)
(134,196)(135,216)(136,218)(137,217)(138,165)(139,167)(140,166)(141,186)
(142,188)(143,187)(144,180)(145,182)(146,181)(147,183)(148,185)(149,184)
(150,177)(151,179)(152,178)(153,171)(154,173)(155,172)(156,174)(157,176)
(158,175)(159,168)(160,170)(161,169)(162,189)(163,191)(164,190);
poly := sub<Sym(218)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope