Polytope of Type {72,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,6}*864a
Also Known As : {72,6|2}. if this polytope has another name.
Group : SmallGroup(864,770)
Rank : 3
Schlafli Type : {72,6}
Number of vertices, edges, etc : 72, 216, 6
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {72,6,2} of size 1728
Vertex Figure Of :
   {2,72,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {36,6}*432a
   3-fold quotients : {72,2}*288, {24,6}*288a
   4-fold quotients : {18,6}*216a
   6-fold quotients : {36,2}*144, {12,6}*144a
   9-fold quotients : {24,2}*96, {8,6}*96
   12-fold quotients : {18,2}*72, {6,6}*72a
   18-fold quotients : {12,2}*48, {4,6}*48a
   24-fold quotients : {9,2}*36
   27-fold quotients : {8,2}*32
   36-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {4,2}*16
   72-fold quotients : {2,3}*12, {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {144,6}*1728a, {72,12}*1728a
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope