Polytope of Type {4,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,6}*1728q
if this polytope has a name.
Group : SmallGroup(1728,47234)
Rank : 4
Schlafli Type : {4,12,6}
Number of vertices, edges, etc : 4, 72, 108, 18
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
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 : {4,12,6}*864c, {2,12,6}*864i
   3-fold quotients : {4,4,6}*576
   4-fold quotients : {2,12,6}*432c
   6-fold quotients : {4,4,6}*288, {2,4,6}*288
   9-fold quotients : {4,12,2}*192a
   12-fold quotients : {2,4,6}*144
   18-fold quotients : {2,12,2}*96, {4,6,2}*96a
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,4,2}*32, {4,2,2}*32
   72-fold quotients : {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, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)
(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)
( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)
( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)
( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)
( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)
( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)
(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)
(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)
(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)
(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)
(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)
(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)
(159,213)(160,214)(161,215)(162,216);;
s1 := (  4,  9)(  5,  7)(  6,  8)( 10, 19)( 11, 20)( 12, 21)( 13, 27)( 14, 25)
( 15, 26)( 16, 23)( 17, 24)( 18, 22)( 31, 36)( 32, 34)( 33, 35)( 37, 46)
( 38, 47)( 39, 48)( 40, 54)( 41, 52)( 42, 53)( 43, 50)( 44, 51)( 45, 49)
( 55, 82)( 56, 83)( 57, 84)( 58, 90)( 59, 88)( 60, 89)( 61, 86)( 62, 87)
( 63, 85)( 64,100)( 65,101)( 66,102)( 67,108)( 68,106)( 69,107)( 70,104)
( 71,105)( 72,103)( 73, 91)( 74, 92)( 75, 93)( 76, 99)( 77, 97)( 78, 98)
( 79, 95)( 80, 96)( 81, 94)(112,117)(113,115)(114,116)(118,127)(119,128)
(120,129)(121,135)(122,133)(123,134)(124,131)(125,132)(126,130)(139,144)
(140,142)(141,143)(145,154)(146,155)(147,156)(148,162)(149,160)(150,161)
(151,158)(152,159)(153,157)(163,190)(164,191)(165,192)(166,198)(167,196)
(168,197)(169,194)(170,195)(171,193)(172,208)(173,209)(174,210)(175,216)
(176,214)(177,215)(178,212)(179,213)(180,211)(181,199)(182,200)(183,201)
(184,207)(185,205)(186,206)(187,203)(188,204)(189,202);;
s2 := (  1, 10)(  2, 17)(  3, 15)(  4, 16)(  5, 14)(  6, 12)(  7, 13)(  8, 11)
(  9, 18)( 20, 26)( 21, 24)( 22, 25)( 28, 37)( 29, 44)( 30, 42)( 31, 43)
( 32, 41)( 33, 39)( 34, 40)( 35, 38)( 36, 45)( 47, 53)( 48, 51)( 49, 52)
( 55, 64)( 56, 71)( 57, 69)( 58, 70)( 59, 68)( 60, 66)( 61, 67)( 62, 65)
( 63, 72)( 74, 80)( 75, 78)( 76, 79)( 82, 91)( 83, 98)( 84, 96)( 85, 97)
( 86, 95)( 87, 93)( 88, 94)( 89, 92)( 90, 99)(101,107)(102,105)(103,106)
(109,118)(110,125)(111,123)(112,124)(113,122)(114,120)(115,121)(116,119)
(117,126)(128,134)(129,132)(130,133)(136,145)(137,152)(138,150)(139,151)
(140,149)(141,147)(142,148)(143,146)(144,153)(155,161)(156,159)(157,160)
(163,172)(164,179)(165,177)(166,178)(167,176)(168,174)(169,175)(170,173)
(171,180)(182,188)(183,186)(184,187)(190,199)(191,206)(192,204)(193,205)
(194,203)(195,201)(196,202)(197,200)(198,207)(209,215)(210,213)(211,214);;
s3 := (  1,110)(  2,109)(  3,111)(  4,116)(  5,115)(  6,117)(  7,113)(  8,112)
(  9,114)( 10,119)( 11,118)( 12,120)( 13,125)( 14,124)( 15,126)( 16,122)
( 17,121)( 18,123)( 19,128)( 20,127)( 21,129)( 22,134)( 23,133)( 24,135)
( 25,131)( 26,130)( 27,132)( 28,137)( 29,136)( 30,138)( 31,143)( 32,142)
( 33,144)( 34,140)( 35,139)( 36,141)( 37,146)( 38,145)( 39,147)( 40,152)
( 41,151)( 42,153)( 43,149)( 44,148)( 45,150)( 46,155)( 47,154)( 48,156)
( 49,161)( 50,160)( 51,162)( 52,158)( 53,157)( 54,159)( 55,164)( 56,163)
( 57,165)( 58,170)( 59,169)( 60,171)( 61,167)( 62,166)( 63,168)( 64,173)
( 65,172)( 66,174)( 67,179)( 68,178)( 69,180)( 70,176)( 71,175)( 72,177)
( 73,182)( 74,181)( 75,183)( 76,188)( 77,187)( 78,189)( 79,185)( 80,184)
( 81,186)( 82,191)( 83,190)( 84,192)( 85,197)( 86,196)( 87,198)( 88,194)
( 89,193)( 90,195)( 91,200)( 92,199)( 93,201)( 94,206)( 95,205)( 96,207)
( 97,203)( 98,202)( 99,204)(100,209)(101,208)(102,210)(103,215)(104,214)
(105,216)(106,212)(107,211)(108,213);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s3*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope