Polytope of Type {2,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,46114)
Rank : 4
Schlafli Type : {2,18,6}
Number of vertices, edges, etc : 2, 72, 216, 24
Order of s₀s₁s₂s₃ : 36
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 : {2,9,6}*864
   3-fold quotients : {2,6,6}*576b
   4-fold quotients : {2,18,6}*432b
   6-fold quotients : {2,3,6}*288
   8-fold quotients : {2,9,6}*216
   9-fold quotients : {2,6,6}*192
   12-fold quotients : {2,18,2}*144, {2,6,6}*144c
   18-fold quotients : {2,3,6}*96, {2,6,3}*96
   24-fold quotients : {2,9,2}*72, {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)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 31)( 16, 33)( 17, 32)
( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)( 25, 36)
( 26, 38)( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)
( 46, 86)( 47, 79)( 48, 81)( 49, 80)( 50, 82)( 51,103)( 52,105)( 53,104)
( 54,106)( 55, 99)( 56,101)( 57,100)( 58,102)( 59,107)( 60,109)( 61,108)
( 62,110)( 63, 91)( 64, 93)( 65, 92)( 66, 94)( 67, 87)( 68, 89)( 69, 88)
( 70, 90)( 71, 95)( 72, 97)( 73, 96)( 74, 98)(112,113)(115,119)(116,121)
(117,120)(118,122)(123,139)(124,141)(125,140)(126,142)(127,135)(128,137)
(129,136)(130,138)(131,143)(132,145)(133,144)(134,146)(147,183)(148,185)
(149,184)(150,186)(151,191)(152,193)(153,192)(154,194)(155,187)(156,189)
(157,188)(158,190)(159,211)(160,213)(161,212)(162,214)(163,207)(164,209)
(165,208)(166,210)(167,215)(168,217)(169,216)(170,218)(171,199)(172,201)
(173,200)(174,202)(175,195)(176,197)(177,196)(178,198)(179,203)(180,205)
(181,204)(182,206);;
s2 := (  3,159)(  4,160)(  5,162)(  6,161)(  7,167)(  8,168)(  9,170)( 10,169)
( 11,163)( 12,164)( 13,166)( 14,165)( 15,147)( 16,148)( 17,150)( 18,149)
( 19,155)( 20,156)( 21,158)( 22,157)( 23,151)( 24,152)( 25,154)( 26,153)
( 27,175)( 28,176)( 29,178)( 30,177)( 31,171)( 32,172)( 33,174)( 34,173)
( 35,179)( 36,180)( 37,182)( 38,181)( 39,123)( 40,124)( 41,126)( 42,125)
( 43,131)( 44,132)( 45,134)( 46,133)( 47,127)( 48,128)( 49,130)( 50,129)
( 51,111)( 52,112)( 53,114)( 54,113)( 55,119)( 56,120)( 57,122)( 58,121)
( 59,115)( 60,116)( 61,118)( 62,117)( 63,139)( 64,140)( 65,142)( 66,141)
( 67,135)( 68,136)( 69,138)( 70,137)( 71,143)( 72,144)( 73,146)( 74,145)
( 75,195)( 76,196)( 77,198)( 78,197)( 79,203)( 80,204)( 81,206)( 82,205)
( 83,199)( 84,200)( 85,202)( 86,201)( 87,183)( 88,184)( 89,186)( 90,185)
( 91,191)( 92,192)( 93,194)( 94,193)( 95,187)( 96,188)( 97,190)( 98,189)
( 99,211)(100,212)(101,214)(102,213)(103,207)(104,208)(105,210)(106,209)
(107,215)(108,216)(109,218)(110,217);;
s3 := (  3,  6)(  7, 10)( 11, 14)( 15, 18)( 19, 22)( 23, 26)( 27, 30)( 31, 34)
( 35, 38)( 39, 78)( 40, 76)( 41, 77)( 42, 75)( 43, 82)( 44, 80)( 45, 81)
( 46, 79)( 47, 86)( 48, 84)( 49, 85)( 50, 83)( 51, 90)( 52, 88)( 53, 89)
( 54, 87)( 55, 94)( 56, 92)( 57, 93)( 58, 91)( 59, 98)( 60, 96)( 61, 97)
( 62, 95)( 63,102)( 64,100)( 65,101)( 66, 99)( 67,106)( 68,104)( 69,105)
( 70,103)( 71,110)( 72,108)( 73,109)( 74,107)(111,114)(115,118)(119,122)
(123,126)(127,130)(131,134)(135,138)(139,142)(143,146)(147,186)(148,184)
(149,185)(150,183)(151,190)(152,188)(153,189)(154,187)(155,194)(156,192)
(157,193)(158,191)(159,198)(160,196)(161,197)(162,195)(163,202)(164,200)
(165,201)(166,199)(167,206)(168,204)(169,205)(170,203)(171,210)(172,208)
(173,209)(174,207)(175,214)(176,212)(177,213)(178,211)(179,218)(180,216)
(181,217)(182,215);;
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*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*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope