Polytope of Type {2,12,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,16947)
Rank : 4
Schlafli Type : {2,12,12}
Number of vertices, edges, etc : 2, 36, 216, 36
Order of s0s1s2s3 : 12
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}*864b, {2,12,6}*864b
3-fold quotients : {2,12,12}*576a
4-fold quotients : {2,6,6}*432b
6-fold quotients : {2,6,12}*288a, {2,12,6}*288a
8-fold quotients : {2,6,6}*216
9-fold quotients : {2,4,12}*192a, {2,12,4}*192a
12-fold quotients : {2,6,6}*144a
18-fold quotients : {2,2,12}*96, {2,12,2}*96, {2,4,6}*96a, {2,6,4}*96a
27-fold quotients : {2,4,4}*64
36-fold quotients : {2,2,6}*48, {2,6,2}*48
54-fold quotients : {2,2,4}*32, {2,4,2}*32
72-fold quotients : {2,2,3}*24, {2,3,2}*24
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, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)( 71, 73)
( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)( 89, 91)
( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109)
(111,138)(112,140)(113,139)(114,144)(115,146)(116,145)(117,141)(118,143)
(119,142)(120,147)(121,149)(122,148)(123,153)(124,155)(125,154)(126,150)
(127,152)(128,151)(129,156)(130,158)(131,157)(132,162)(133,164)(134,163)
(135,159)(136,161)(137,160)(165,192)(166,194)(167,193)(168,198)(169,200)
(170,199)(171,195)(172,197)(173,196)(174,201)(175,203)(176,202)(177,207)
(178,209)(179,208)(180,204)(181,206)(182,205)(183,210)(184,212)(185,211)
(186,216)(187,218)(188,217)(189,213)(190,215)(191,214);;
s2 := ( 3,114)( 4,115)( 5,116)( 6,111)( 7,112)( 8,113)( 9,117)( 10,118)
( 11,119)( 12,132)( 13,133)( 14,134)( 15,129)( 16,130)( 17,131)( 18,135)
( 19,136)( 20,137)( 21,123)( 22,124)( 23,125)( 24,120)( 25,121)( 26,122)
( 27,126)( 28,127)( 29,128)( 30,141)( 31,142)( 32,143)( 33,138)( 34,139)
( 35,140)( 36,144)( 37,145)( 38,146)( 39,159)( 40,160)( 41,161)( 42,156)
( 43,157)( 44,158)( 45,162)( 46,163)( 47,164)( 48,150)( 49,151)( 50,152)
( 51,147)( 52,148)( 53,149)( 54,153)( 55,154)( 56,155)( 57,168)( 58,169)
( 59,170)( 60,165)( 61,166)( 62,167)( 63,171)( 64,172)( 65,173)( 66,186)
( 67,187)( 68,188)( 69,183)( 70,184)( 71,185)( 72,189)( 73,190)( 74,191)
( 75,177)( 76,178)( 77,179)( 78,174)( 79,175)( 80,176)( 81,180)( 82,181)
( 83,182)( 84,195)( 85,196)( 86,197)( 87,192)( 88,193)( 89,194)( 90,198)
( 91,199)( 92,200)( 93,213)( 94,214)( 95,215)( 96,210)( 97,211)( 98,212)
( 99,216)(100,217)(101,218)(102,204)(103,205)(104,206)(105,201)(106,202)
(107,203)(108,207)(109,208)(110,209);;
s3 := ( 3, 12)( 4, 14)( 5, 13)( 6, 16)( 7, 15)( 8, 17)( 9, 20)( 10, 19)
( 11, 18)( 22, 23)( 24, 25)( 27, 29)( 30, 39)( 31, 41)( 32, 40)( 33, 43)
( 34, 42)( 35, 44)( 36, 47)( 37, 46)( 38, 45)( 49, 50)( 51, 52)( 54, 56)
( 57, 66)( 58, 68)( 59, 67)( 60, 70)( 61, 69)( 62, 71)( 63, 74)( 64, 73)
( 65, 72)( 76, 77)( 78, 79)( 81, 83)( 84, 93)( 85, 95)( 86, 94)( 87, 97)
( 88, 96)( 89, 98)( 90,101)( 91,100)( 92, 99)(103,104)(105,106)(108,110)
(111,174)(112,176)(113,175)(114,178)(115,177)(116,179)(117,182)(118,181)
(119,180)(120,165)(121,167)(122,166)(123,169)(124,168)(125,170)(126,173)
(127,172)(128,171)(129,183)(130,185)(131,184)(132,187)(133,186)(134,188)
(135,191)(136,190)(137,189)(138,201)(139,203)(140,202)(141,205)(142,204)
(143,206)(144,209)(145,208)(146,207)(147,192)(148,194)(149,193)(150,196)
(151,195)(152,197)(153,200)(154,199)(155,198)(156,210)(157,212)(158,211)
(159,214)(160,213)(161,215)(162,218)(163,217)(164,216);;
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,
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s3*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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)
( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)
( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)
(107,109)(111,138)(112,140)(113,139)(114,144)(115,146)(116,145)(117,141)
(118,143)(119,142)(120,147)(121,149)(122,148)(123,153)(124,155)(125,154)
(126,150)(127,152)(128,151)(129,156)(130,158)(131,157)(132,162)(133,164)
(134,163)(135,159)(136,161)(137,160)(165,192)(166,194)(167,193)(168,198)
(169,200)(170,199)(171,195)(172,197)(173,196)(174,201)(175,203)(176,202)
(177,207)(178,209)(179,208)(180,204)(181,206)(182,205)(183,210)(184,212)
(185,211)(186,216)(187,218)(188,217)(189,213)(190,215)(191,214);
s2 := Sym(218)!( 3,114)( 4,115)( 5,116)( 6,111)( 7,112)( 8,113)( 9,117)
( 10,118)( 11,119)( 12,132)( 13,133)( 14,134)( 15,129)( 16,130)( 17,131)
( 18,135)( 19,136)( 20,137)( 21,123)( 22,124)( 23,125)( 24,120)( 25,121)
( 26,122)( 27,126)( 28,127)( 29,128)( 30,141)( 31,142)( 32,143)( 33,138)
( 34,139)( 35,140)( 36,144)( 37,145)( 38,146)( 39,159)( 40,160)( 41,161)
( 42,156)( 43,157)( 44,158)( 45,162)( 46,163)( 47,164)( 48,150)( 49,151)
( 50,152)( 51,147)( 52,148)( 53,149)( 54,153)( 55,154)( 56,155)( 57,168)
( 58,169)( 59,170)( 60,165)( 61,166)( 62,167)( 63,171)( 64,172)( 65,173)
( 66,186)( 67,187)( 68,188)( 69,183)( 70,184)( 71,185)( 72,189)( 73,190)
( 74,191)( 75,177)( 76,178)( 77,179)( 78,174)( 79,175)( 80,176)( 81,180)
( 82,181)( 83,182)( 84,195)( 85,196)( 86,197)( 87,192)( 88,193)( 89,194)
( 90,198)( 91,199)( 92,200)( 93,213)( 94,214)( 95,215)( 96,210)( 97,211)
( 98,212)( 99,216)(100,217)(101,218)(102,204)(103,205)(104,206)(105,201)
(106,202)(107,203)(108,207)(109,208)(110,209);
s3 := Sym(218)!( 3, 12)( 4, 14)( 5, 13)( 6, 16)( 7, 15)( 8, 17)( 9, 20)
( 10, 19)( 11, 18)( 22, 23)( 24, 25)( 27, 29)( 30, 39)( 31, 41)( 32, 40)
( 33, 43)( 34, 42)( 35, 44)( 36, 47)( 37, 46)( 38, 45)( 49, 50)( 51, 52)
( 54, 56)( 57, 66)( 58, 68)( 59, 67)( 60, 70)( 61, 69)( 62, 71)( 63, 74)
( 64, 73)( 65, 72)( 76, 77)( 78, 79)( 81, 83)( 84, 93)( 85, 95)( 86, 94)
( 87, 97)( 88, 96)( 89, 98)( 90,101)( 91,100)( 92, 99)(103,104)(105,106)
(108,110)(111,174)(112,176)(113,175)(114,178)(115,177)(116,179)(117,182)
(118,181)(119,180)(120,165)(121,167)(122,166)(123,169)(124,168)(125,170)
(126,173)(127,172)(128,171)(129,183)(130,185)(131,184)(132,187)(133,186)
(134,188)(135,191)(136,190)(137,189)(138,201)(139,203)(140,202)(141,205)
(142,204)(143,206)(144,209)(145,208)(146,207)(147,192)(148,194)(149,193)
(150,196)(151,195)(152,197)(153,200)(154,199)(155,198)(156,210)(157,212)
(158,211)(159,214)(160,213)(161,215)(162,218)(163,217)(164,216);
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, s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s3*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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
to this polytope