Polytope of Type {2,6,6}

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

to this polytope