Polytope of Type {6,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,6}*1728h
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {6,12,6}
Number of vertices, edges, etc : 6, 72, 72, 12
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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) :
   3-fold quotients : {6,4,6}*576a
   4-fold quotients : {6,6,6}*432g
   6-fold quotients : {6,4,3}*288
   8-fold quotients : {3,6,6}*216b
   9-fold quotients : {2,4,6}*192
   12-fold quotients : {6,2,6}*144, {6,6,2}*144c
   18-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
   24-fold quotients : {3,2,6}*72, {3,6,2}*72, {6,2,3}*72
   36-fold quotients : {2,4,3}*48, {2,2,6}*48, {6,2,2}*48
   48-fold quotients : {3,2,3}*36
   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 := (  5,  9)(  6, 10)(  7, 11)(  8, 12)( 13, 25)( 14, 26)( 15, 27)( 16, 28)
( 17, 33)( 18, 34)( 19, 35)( 20, 36)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 69)( 54, 70)( 55, 71)( 56, 72)( 57, 65)( 58, 66)( 59, 67)( 60, 68)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85, 97)( 86, 98)( 87, 99)( 88,100)
( 89,105)( 90,106)( 91,107)( 92,108)( 93,101)( 94,102)( 95,103)( 96,104)
(113,117)(114,118)(115,119)(116,120)(121,133)(122,134)(123,135)(124,136)
(125,141)(126,142)(127,143)(128,144)(129,137)(130,138)(131,139)(132,140)
(149,153)(150,154)(151,155)(152,156)(157,169)(158,170)(159,171)(160,172)
(161,177)(162,178)(163,179)(164,180)(165,173)(166,174)(167,175)(168,176)
(185,189)(186,190)(187,191)(188,192)(193,205)(194,206)(195,207)(196,208)
(197,213)(198,214)(199,215)(200,216)(201,209)(202,210)(203,211)(204,212);;
s1 := (  1,127)(  2,128)(  3,125)(  4,126)(  5,123)(  6,124)(  7,121)(  8,122)
(  9,131)( 10,132)( 11,129)( 12,130)( 13,115)( 14,116)( 15,113)( 16,114)
( 17,111)( 18,112)( 19,109)( 20,110)( 21,119)( 22,120)( 23,117)( 24,118)
( 25,139)( 26,140)( 27,137)( 28,138)( 29,135)( 30,136)( 31,133)( 32,134)
( 33,143)( 34,144)( 35,141)( 36,142)( 37,163)( 38,164)( 39,161)( 40,162)
( 41,159)( 42,160)( 43,157)( 44,158)( 45,167)( 46,168)( 47,165)( 48,166)
( 49,151)( 50,152)( 51,149)( 52,150)( 53,147)( 54,148)( 55,145)( 56,146)
( 57,155)( 58,156)( 59,153)( 60,154)( 61,175)( 62,176)( 63,173)( 64,174)
( 65,171)( 66,172)( 67,169)( 68,170)( 69,179)( 70,180)( 71,177)( 72,178)
( 73,199)( 74,200)( 75,197)( 76,198)( 77,195)( 78,196)( 79,193)( 80,194)
( 81,203)( 82,204)( 83,201)( 84,202)( 85,187)( 86,188)( 87,185)( 88,186)
( 89,183)( 90,184)( 91,181)( 92,182)( 93,191)( 94,192)( 95,189)( 96,190)
( 97,211)( 98,212)( 99,209)(100,210)(101,207)(102,208)(103,205)(104,206)
(105,215)(106,216)(107,213)(108,214);;
s2 := (  3,  4)(  7,  8)( 11, 12)( 13, 25)( 14, 26)( 15, 28)( 16, 27)( 17, 29)
( 18, 30)( 19, 32)( 20, 31)( 21, 33)( 22, 34)( 23, 36)( 24, 35)( 37, 73)
( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 81)
( 46, 82)( 47, 84)( 48, 83)( 49, 97)( 50, 98)( 51,100)( 52, 99)( 53,101)
( 54,102)( 55,104)( 56,103)( 57,105)( 58,106)( 59,108)( 60,107)( 61, 85)
( 62, 86)( 63, 88)( 64, 87)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 93)
( 70, 94)( 71, 96)( 72, 95)(111,112)(115,116)(119,120)(121,133)(122,134)
(123,136)(124,135)(125,137)(126,138)(127,140)(128,139)(129,141)(130,142)
(131,144)(132,143)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)
(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,205)(158,206)
(159,208)(160,207)(161,209)(162,210)(163,212)(164,211)(165,213)(166,214)
(167,216)(168,215)(169,193)(170,194)(171,196)(172,195)(173,197)(174,198)
(175,200)(176,199)(177,201)(178,202)(179,204)(180,203);;
s3 := (  1, 37)(  2, 40)(  3, 39)(  4, 38)(  5, 41)(  6, 44)(  7, 43)(  8, 42)
(  9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)( 24, 58)
( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 65)( 30, 68)( 31, 67)( 32, 66)
( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 74, 76)( 78, 80)( 82, 84)( 86, 88)
( 90, 92)( 94, 96)( 98,100)(102,104)(106,108)(109,145)(110,148)(111,147)
(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)
(120,154)(121,157)(122,160)(123,159)(124,158)(125,161)(126,164)(127,163)
(128,162)(129,165)(130,168)(131,167)(132,166)(133,169)(134,172)(135,171)
(136,170)(137,173)(138,176)(139,175)(140,174)(141,177)(142,180)(143,179)
(144,178)(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)(206,208)
(210,212)(214,216);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope