Polytope of Type {2,12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,12}*1728i
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 : {2,12,6}*864h
   3-fold quotients : {2,4,12}*576
   6-fold quotients : {2,4,6}*288
   12-fold quotients : {2,4,6}*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 := (  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);;
s2 := (  3,112)(  4,111)(  5,113)(  6,115)(  7,114)(  8,116)(  9,118)( 10,117)
( 11,119)( 12,136)( 13,135)( 14,137)( 15,130)( 16,129)( 17,131)( 18,133)
( 19,132)( 20,134)( 21,124)( 22,123)( 23,125)( 24,127)( 25,126)( 26,128)
( 27,121)( 28,120)( 29,122)( 30,139)( 31,138)( 32,140)( 33,142)( 34,141)
( 35,143)( 36,145)( 37,144)( 38,146)( 39,163)( 40,162)( 41,164)( 42,157)
( 43,156)( 44,158)( 45,160)( 46,159)( 47,161)( 48,151)( 49,150)( 50,152)
( 51,154)( 52,153)( 53,155)( 54,148)( 55,147)( 56,149)( 57,166)( 58,165)
( 59,167)( 60,169)( 61,168)( 62,170)( 63,172)( 64,171)( 65,173)( 66,190)
( 67,189)( 68,191)( 69,184)( 70,183)( 71,185)( 72,187)( 73,186)( 74,188)
( 75,178)( 76,177)( 77,179)( 78,181)( 79,180)( 80,182)( 81,175)( 82,174)
( 83,176)( 84,193)( 85,192)( 86,194)( 87,196)( 88,195)( 89,197)( 90,199)
( 91,198)( 92,200)( 93,217)( 94,216)( 95,218)( 96,211)( 97,210)( 98,212)
( 99,214)(100,213)(101,215)(102,205)(103,204)(104,206)(105,208)(106,207)
(107,209)(108,202)(109,201)(110,203);;
s3 := (  3, 69)(  4, 71)(  5, 70)(  6, 66)(  7, 68)(  8, 67)(  9, 72)( 10, 74)
( 11, 73)( 12, 60)( 13, 62)( 14, 61)( 15, 57)( 16, 59)( 17, 58)( 18, 63)
( 19, 65)( 20, 64)( 21, 78)( 22, 80)( 23, 79)( 24, 75)( 25, 77)( 26, 76)
( 27, 81)( 28, 83)( 29, 82)( 30, 96)( 31, 98)( 32, 97)( 33, 93)( 34, 95)
( 35, 94)( 36, 99)( 37,101)( 38,100)( 39, 87)( 40, 89)( 41, 88)( 42, 84)
( 43, 86)( 44, 85)( 45, 90)( 46, 92)( 47, 91)( 48,105)( 49,107)( 50,106)
( 51,102)( 52,104)( 53,103)( 54,108)( 55,110)( 56,109)(111,204)(112,206)
(113,205)(114,201)(115,203)(116,202)(117,207)(118,209)(119,208)(120,195)
(121,197)(122,196)(123,192)(124,194)(125,193)(126,198)(127,200)(128,199)
(129,213)(130,215)(131,214)(132,210)(133,212)(134,211)(135,216)(136,218)
(137,217)(138,177)(139,179)(140,178)(141,174)(142,176)(143,175)(144,180)
(145,182)(146,181)(147,168)(148,170)(149,169)(150,165)(151,167)(152,166)
(153,171)(154,173)(155,172)(156,186)(157,188)(158,187)(159,183)(160,185)
(161,184)(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*s1*s2*s1*s2*s3*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*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope