Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1728b
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) :
   3-fold quotients : {6,6,6}*576a, {2,6,6}*576b
   4-fold quotients : {6,6,6}*432e
   6-fold quotients : {2,3,6}*288, {6,6,3}*288
   9-fold quotients : {2,6,6}*192
   12-fold quotients : {2,6,6}*144c, {6,6,2}*144a
   18-fold quotients : {2,3,6}*96, {2,6,3}*96
   24-fold quotients : {2,3,6}*72
   36-fold quotients : {2,3,3}*48, {2,6,2}*48, {6,2,2}*48
   72-fold quotients : {2,3,2}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s2*s1*s3*s2*s3*s2*s1*s3*s2*s3> of order 2.
      12 facets:
         12 of {6,6}*72a
      6 vertex figures:
         6 of 2-fold non-regular quotient of {6,6}*288b
   P/N, where N=<s2*s1*s2*s3*s2*s3*s2*s1*s3*s2> of order 2.
      12 facets:
         12 of {6,6}*72a
      6 vertex figures:
         6 of 2-fold non-regular quotient of {6,6}*288b
   P/N, where N=<s1*s3*s2*s3*s2*s1*s2*s3*s2*s3> of order 3.
      8 facets:
         8 of {6,6}*72a
      6 vertex figures:
         6 of 3-fold non-regular quotient of {6,6}*288b
   P/N, where N=<s1*s2*s3*s2*s1*s3*s2*s3*s2> of order 4.
      6 facets:
         6 of {6,6}*72a
      6 vertex figures:
         6 of 4-fold non-regular quotient of {6,6}*288b

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

References : None.
to this polytope