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Polytope of Type {2,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 4
Schlafli Type : {2,6,6}
Number of vertices, edges, etc : 2, 72, 216, 72
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   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 : {2,3,6}*864
   3-fold quotients : {2,6,6}*576b
   4-fold quotients : {2,6,6}*432c
   6-fold quotients : {2,3,6}*288
   8-fold quotients : {2,3,6}*216
   9-fold quotients : {2,6,6}*192
   12-fold quotients : {2,6,6}*144c
   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
   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 := (1,2);;
s1 := (  4,  5)(  8,  9)( 12, 13)( 15, 27)( 16, 29)( 17, 28)( 18, 30)( 19, 31)
( 20, 33)( 21, 32)( 22, 34)( 23, 35)( 24, 37)( 25, 36)( 26, 38)( 39, 75)
( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 82)( 47, 83)
( 48, 85)( 49, 84)( 50, 86)( 51, 99)( 52,101)( 53,100)( 54,102)( 55,103)
( 56,105)( 57,104)( 58,106)( 59,107)( 60,109)( 61,108)( 62,110)( 63, 87)
( 64, 89)( 65, 88)( 66, 90)( 67, 91)( 68, 93)( 69, 92)( 70, 94)( 71, 95)
( 72, 97)( 73, 96)( 74, 98)(112,113)(116,117)(120,121)(123,135)(124,137)
(125,136)(126,138)(127,139)(128,141)(129,140)(130,142)(131,143)(132,145)
(133,144)(134,146)(147,183)(148,185)(149,184)(150,186)(151,187)(152,189)
(153,188)(154,190)(155,191)(156,193)(157,192)(158,194)(159,207)(160,209)
(161,208)(162,210)(163,211)(164,213)(165,212)(166,214)(167,215)(168,217)
(169,216)(170,218)(171,195)(172,197)(173,196)(174,198)(175,199)(176,201)
(177,200)(178,202)(179,203)(180,205)(181,204)(182,206);;
s2 := (  3,199)(  4,202)(  5,201)(  6,200)(  7,203)(  8,206)(  9,205)( 10,204)
( 11,195)( 12,198)( 13,197)( 14,196)( 15,191)( 16,194)( 17,193)( 18,192)
( 19,183)( 20,186)( 21,185)( 22,184)( 23,187)( 24,190)( 25,189)( 26,188)
( 27,207)( 28,210)( 29,209)( 30,208)( 31,211)( 32,214)( 33,213)( 34,212)
( 35,215)( 36,218)( 37,217)( 38,216)( 39,163)( 40,166)( 41,165)( 42,164)
( 43,167)( 44,170)( 45,169)( 46,168)( 47,159)( 48,162)( 49,161)( 50,160)
( 51,155)( 52,158)( 53,157)( 54,156)( 55,147)( 56,150)( 57,149)( 58,148)
( 59,151)( 60,154)( 61,153)( 62,152)( 63,171)( 64,174)( 65,173)( 66,172)
( 67,175)( 68,178)( 69,177)( 70,176)( 71,179)( 72,182)( 73,181)( 74,180)
( 75,127)( 76,130)( 77,129)( 78,128)( 79,131)( 80,134)( 81,133)( 82,132)
( 83,123)( 84,126)( 85,125)( 86,124)( 87,119)( 88,122)( 89,121)( 90,120)
( 91,111)( 92,114)( 93,113)( 94,112)( 95,115)( 96,118)( 97,117)( 98,116)
( 99,135)(100,138)(101,137)(102,136)(103,139)(104,142)(105,141)(106,140)
(107,143)(108,146)(109,145)(110,144);;
s3 := (  3,  6)(  7, 14)(  8, 12)(  9, 13)( 10, 11)( 15, 18)( 19, 26)( 20, 24)
( 21, 25)( 22, 23)( 27, 30)( 31, 38)( 32, 36)( 33, 37)( 34, 35)( 39, 78)
( 40, 76)( 41, 77)( 42, 75)( 43, 86)( 44, 84)( 45, 85)( 46, 83)( 47, 82)
( 48, 80)( 49, 81)( 50, 79)( 51, 90)( 52, 88)( 53, 89)( 54, 87)( 55, 98)
( 56, 96)( 57, 97)( 58, 95)( 59, 94)( 60, 92)( 61, 93)( 62, 91)( 63,102)
( 64,100)( 65,101)( 66, 99)( 67,110)( 68,108)( 69,109)( 70,107)( 71,106)
( 72,104)( 73,105)( 74,103)(111,114)(115,122)(116,120)(117,121)(118,119)
(123,126)(127,134)(128,132)(129,133)(130,131)(135,138)(139,146)(140,144)
(141,145)(142,143)(147,186)(148,184)(149,185)(150,183)(151,194)(152,192)
(153,193)(154,191)(155,190)(156,188)(157,189)(158,187)(159,198)(160,196)
(161,197)(162,195)(163,206)(164,204)(165,205)(166,203)(167,202)(168,200)
(169,201)(170,199)(171,210)(172,208)(173,209)(174,207)(175,218)(176,216)
(177,217)(178,215)(179,214)(180,212)(181,213)(182,211);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  4,  5)(  8,  9)( 12, 13)( 15, 27)( 16, 29)( 17, 28)( 18, 30)
( 19, 31)( 20, 33)( 21, 32)( 22, 34)( 23, 35)( 24, 37)( 25, 36)( 26, 38)
( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 82)
( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51, 99)( 52,101)( 53,100)( 54,102)
( 55,103)( 56,105)( 57,104)( 58,106)( 59,107)( 60,109)( 61,108)( 62,110)
( 63, 87)( 64, 89)( 65, 88)( 66, 90)( 67, 91)( 68, 93)( 69, 92)( 70, 94)
( 71, 95)( 72, 97)( 73, 96)( 74, 98)(112,113)(116,117)(120,121)(123,135)
(124,137)(125,136)(126,138)(127,139)(128,141)(129,140)(130,142)(131,143)
(132,145)(133,144)(134,146)(147,183)(148,185)(149,184)(150,186)(151,187)
(152,189)(153,188)(154,190)(155,191)(156,193)(157,192)(158,194)(159,207)
(160,209)(161,208)(162,210)(163,211)(164,213)(165,212)(166,214)(167,215)
(168,217)(169,216)(170,218)(171,195)(172,197)(173,196)(174,198)(175,199)
(176,201)(177,200)(178,202)(179,203)(180,205)(181,204)(182,206);
s2 := Sym(218)!(  3,199)(  4,202)(  5,201)(  6,200)(  7,203)(  8,206)(  9,205)
( 10,204)( 11,195)( 12,198)( 13,197)( 14,196)( 15,191)( 16,194)( 17,193)
( 18,192)( 19,183)( 20,186)( 21,185)( 22,184)( 23,187)( 24,190)( 25,189)
( 26,188)( 27,207)( 28,210)( 29,209)( 30,208)( 31,211)( 32,214)( 33,213)
( 34,212)( 35,215)( 36,218)( 37,217)( 38,216)( 39,163)( 40,166)( 41,165)
( 42,164)( 43,167)( 44,170)( 45,169)( 46,168)( 47,159)( 48,162)( 49,161)
( 50,160)( 51,155)( 52,158)( 53,157)( 54,156)( 55,147)( 56,150)( 57,149)
( 58,148)( 59,151)( 60,154)( 61,153)( 62,152)( 63,171)( 64,174)( 65,173)
( 66,172)( 67,175)( 68,178)( 69,177)( 70,176)( 71,179)( 72,182)( 73,181)
( 74,180)( 75,127)( 76,130)( 77,129)( 78,128)( 79,131)( 80,134)( 81,133)
( 82,132)( 83,123)( 84,126)( 85,125)( 86,124)( 87,119)( 88,122)( 89,121)
( 90,120)( 91,111)( 92,114)( 93,113)( 94,112)( 95,115)( 96,118)( 97,117)
( 98,116)( 99,135)(100,138)(101,137)(102,136)(103,139)(104,142)(105,141)
(106,140)(107,143)(108,146)(109,145)(110,144);
s3 := Sym(218)!(  3,  6)(  7, 14)(  8, 12)(  9, 13)( 10, 11)( 15, 18)( 19, 26)
( 20, 24)( 21, 25)( 22, 23)( 27, 30)( 31, 38)( 32, 36)( 33, 37)( 34, 35)
( 39, 78)( 40, 76)( 41, 77)( 42, 75)( 43, 86)( 44, 84)( 45, 85)( 46, 83)
( 47, 82)( 48, 80)( 49, 81)( 50, 79)( 51, 90)( 52, 88)( 53, 89)( 54, 87)
( 55, 98)( 56, 96)( 57, 97)( 58, 95)( 59, 94)( 60, 92)( 61, 93)( 62, 91)
( 63,102)( 64,100)( 65,101)( 66, 99)( 67,110)( 68,108)( 69,109)( 70,107)
( 71,106)( 72,104)( 73,105)( 74,103)(111,114)(115,122)(116,120)(117,121)
(118,119)(123,126)(127,134)(128,132)(129,133)(130,131)(135,138)(139,146)
(140,144)(141,145)(142,143)(147,186)(148,184)(149,185)(150,183)(151,194)
(152,192)(153,193)(154,191)(155,190)(156,188)(157,189)(158,187)(159,198)
(160,196)(161,197)(162,195)(163,206)(164,204)(165,205)(166,203)(167,202)
(168,200)(169,201)(170,199)(171,210)(172,208)(173,209)(174,207)(175,218)
(176,216)(177,217)(178,215)(179,214)(180,212)(181,213)(182,211);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >; 
 

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