Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 6, 72, 72, 24
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 : {6,6,3}*864
   3-fold quotients : {6,6,6}*576a, {2,6,6}*576a
   4-fold quotients : {6,6,6}*432c
   6-fold quotients : {2,6,3}*288, {6,6,3}*288
   8-fold quotients : {6,6,3}*216b
   9-fold quotients : {2,6,6}*192
   12-fold quotients : {2,6,6}*144b, {6,2,6}*144
   18-fold quotients : {2,3,6}*96, {2,6,3}*96
   24-fold quotients : {2,6,3}*72, {3,2,6}*72, {6,2,3}*72
   36-fold quotients : {2,3,3}*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 := ( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)
( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)
( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)(124,136)
(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)
(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)
(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)(196,208)
(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 17)(  6, 18)(  7, 20)(  8, 19)
(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 27, 28)( 31, 32)( 35, 36)( 37, 85)
( 38, 86)( 39, 88)( 40, 87)( 41, 89)( 42, 90)( 43, 92)( 44, 91)( 45, 93)
( 46, 94)( 47, 96)( 48, 95)( 49, 73)( 50, 74)( 51, 76)( 52, 75)( 53, 77)
( 54, 78)( 55, 80)( 56, 79)( 57, 81)( 58, 82)( 59, 84)( 60, 83)( 61, 97)
( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)( 69,105)
( 70,106)( 71,108)( 72,107)(109,121)(110,122)(111,124)(112,123)(113,125)
(114,126)(115,128)(116,127)(117,129)(118,130)(119,132)(120,131)(135,136)
(139,140)(143,144)(145,193)(146,194)(147,196)(148,195)(149,197)(150,198)
(151,200)(152,199)(153,201)(154,202)(155,204)(156,203)(157,181)(158,182)
(159,184)(160,183)(161,185)(162,186)(163,188)(164,187)(165,189)(166,190)
(167,192)(168,191)(169,205)(170,206)(171,208)(172,207)(173,209)(174,210)
(175,212)(176,211)(177,213)(178,214)(179,216)(180,215);;
s2 := (  1, 37)(  2, 40)(  3, 39)(  4, 38)(  5, 45)(  6, 48)(  7, 47)(  8, 46)
(  9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 57)( 18, 60)( 19, 59)( 20, 58)( 21, 53)( 22, 56)( 23, 55)( 24, 54)
( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 69)( 30, 72)( 31, 71)( 32, 70)
( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 74, 76)( 77, 81)( 78, 84)( 79, 83)
( 80, 82)( 86, 88)( 89, 93)( 90, 96)( 91, 95)( 92, 94)( 98,100)(101,105)
(102,108)(103,107)(104,106)(109,145)(110,148)(111,147)(112,146)(113,153)
(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)(120,150)(121,157)
(122,160)(123,159)(124,158)(125,165)(126,168)(127,167)(128,166)(129,161)
(130,164)(131,163)(132,162)(133,169)(134,172)(135,171)(136,170)(137,177)
(138,180)(139,179)(140,178)(141,173)(142,176)(143,175)(144,174)(182,184)
(185,189)(186,192)(187,191)(188,190)(194,196)(197,201)(198,204)(199,203)
(200,202)(206,208)(209,213)(210,216)(211,215)(212,214);;
s3 := (  1,114)(  2,113)(  3,115)(  4,116)(  5,110)(  6,109)(  7,111)(  8,112)
(  9,118)( 10,117)( 11,119)( 12,120)( 13,126)( 14,125)( 15,127)( 16,128)
( 17,122)( 18,121)( 19,123)( 20,124)( 21,130)( 22,129)( 23,131)( 24,132)
( 25,138)( 26,137)( 27,139)( 28,140)( 29,134)( 30,133)( 31,135)( 32,136)
( 33,142)( 34,141)( 35,143)( 36,144)( 37,186)( 38,185)( 39,187)( 40,188)
( 41,182)( 42,181)( 43,183)( 44,184)( 45,190)( 46,189)( 47,191)( 48,192)
( 49,198)( 50,197)( 51,199)( 52,200)( 53,194)( 54,193)( 55,195)( 56,196)
( 57,202)( 58,201)( 59,203)( 60,204)( 61,210)( 62,209)( 63,211)( 64,212)
( 65,206)( 66,205)( 67,207)( 68,208)( 69,214)( 70,213)( 71,215)( 72,216)
( 73,150)( 74,149)( 75,151)( 76,152)( 77,146)( 78,145)( 79,147)( 80,148)
( 81,154)( 82,153)( 83,155)( 84,156)( 85,162)( 86,161)( 87,163)( 88,164)
( 89,158)( 90,157)( 91,159)( 92,160)( 93,166)( 94,165)( 95,167)( 96,168)
( 97,174)( 98,173)( 99,175)(100,176)(101,170)(102,169)(103,171)(104,172)
(105,178)(106,177)(107,179)(108,180);;
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*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope