Polytope of Type {12,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,14771)
Rank : 4
Schlafli Type : {12,12,6}
Number of vertices, edges, etc : 12, 72, 36, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   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 : {12,6,6}*864a, {6,12,6}*864a
   3-fold quotients : {12,4,6}*576
   4-fold quotients : {12,6,3}*432a, {6,6,6}*432a
   6-fold quotients : {12,2,6}*288, {6,4,6}*288
   8-fold quotients : {3,6,6}*216a, {6,6,3}*216a
   9-fold quotients : {12,4,2}*192a, {4,4,6}*192
   12-fold quotients : {12,2,3}*144, {6,2,6}*144
   16-fold quotients : {3,6,3}*108
   18-fold quotients : {12,2,2}*96, {2,4,6}*96a, {4,2,6}*96, {6,4,2}*96a
   24-fold quotients : {3,2,6}*72, {6,2,3}*72
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48, {6,2,2}*48
   48-fold quotients : {3,2,3}*36
   54-fold quotients : {2,4,2}*32, {4,2,2}*32
   72-fold quotients : {2,2,3}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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