Polytope of Type {2,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,9}*1944e
if this polytope has a name.
Group : SmallGroup(1944,952)
Rank : 4
Schlafli Type : {2,6,9}
Number of vertices, edges, etc : 2, 54, 243, 81
Order of s₀s₁s₂s₃ : 18
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) :
   3-fold quotients : {2,6,9}*648a, {2,6,9}*648c, {2,6,9}*648d, {2,6,3}*648
   9-fold quotients : {2,6,9}*216, {2,6,3}*216
   27-fold quotients : {2,2,9}*72, {2,6,3}*72
   81-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  6, 10)(  7, 11)(  8,  9)( 15, 19)( 16, 20)( 17, 18)( 24, 28)( 25, 29)
( 26, 27)( 30, 57)( 31, 58)( 32, 59)( 33, 64)( 34, 65)( 35, 63)( 36, 62)
( 37, 60)( 38, 61)( 39, 66)( 40, 67)( 41, 68)( 42, 73)( 43, 74)( 44, 72)
( 45, 71)( 46, 69)( 47, 70)( 48, 75)( 49, 76)( 50, 77)( 51, 82)( 52, 83)
( 53, 81)( 54, 80)( 55, 78)( 56, 79)( 87, 91)( 88, 92)( 89, 90)( 96,100)
( 97,101)( 98, 99)(105,109)(106,110)(107,108)(111,138)(112,139)(113,140)
(114,145)(115,146)(116,144)(117,143)(118,141)(119,142)(120,147)(121,148)
(122,149)(123,154)(124,155)(125,153)(126,152)(127,150)(128,151)(129,156)
(130,157)(131,158)(132,163)(133,164)(134,162)(135,161)(136,159)(137,160)
(168,172)(169,173)(170,171)(177,181)(178,182)(179,180)(186,190)(187,191)
(188,189)(192,219)(193,220)(194,221)(195,226)(196,227)(197,225)(198,224)
(199,222)(200,223)(201,228)(202,229)(203,230)(204,235)(205,236)(206,234)
(207,233)(208,231)(209,232)(210,237)(211,238)(212,239)(213,244)(214,245)
(215,243)(216,242)(217,240)(218,241);;
s2 := (  3, 30)(  4, 32)(  5, 31)(  6, 33)(  7, 35)(  8, 34)(  9, 36)( 10, 38)
( 11, 37)( 12, 48)( 13, 50)( 14, 49)( 15, 51)( 16, 53)( 17, 52)( 18, 54)
( 19, 56)( 20, 55)( 21, 39)( 22, 41)( 23, 40)( 24, 42)( 25, 44)( 26, 43)
( 27, 45)( 28, 47)( 29, 46)( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)
( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 84,210)
( 85,212)( 86,211)( 87,213)( 88,215)( 89,214)( 90,216)( 91,218)( 92,217)
( 93,201)( 94,203)( 95,202)( 96,204)( 97,206)( 98,205)( 99,207)(100,209)
(101,208)(102,192)(103,194)(104,193)(105,195)(106,197)(107,196)(108,198)
(109,200)(110,199)(111,183)(112,185)(113,184)(114,186)(115,188)(116,187)
(117,189)(118,191)(119,190)(120,174)(121,176)(122,175)(123,177)(124,179)
(125,178)(126,180)(127,182)(128,181)(129,165)(130,167)(131,166)(132,168)
(133,170)(134,169)(135,171)(136,173)(137,172)(138,237)(139,239)(140,238)
(141,240)(142,242)(143,241)(144,243)(145,245)(146,244)(147,228)(148,230)
(149,229)(150,231)(151,233)(152,232)(153,234)(154,236)(155,235)(156,219)
(157,221)(158,220)(159,222)(160,224)(161,223)(162,225)(163,227)(164,226);;
s3 := (  3, 84)(  4, 86)(  5, 85)(  6, 89)(  7, 88)(  8, 87)(  9, 91)( 10, 90)
( 11, 92)( 12,102)( 13,104)( 14,103)( 15,107)( 16,106)( 17,105)( 18,109)
( 19,108)( 20,110)( 21, 93)( 22, 95)( 23, 94)( 24, 98)( 25, 97)( 26, 96)
( 27,100)( 28, 99)( 29,101)( 30,145)( 31,144)( 32,146)( 33,138)( 34,140)
( 35,139)( 36,143)( 37,142)( 38,141)( 39,163)( 40,162)( 41,164)( 42,156)
( 43,158)( 44,157)( 45,161)( 46,160)( 47,159)( 48,154)( 49,153)( 50,155)
( 51,147)( 52,149)( 53,148)( 54,152)( 55,151)( 56,150)( 57,114)( 58,116)
( 59,115)( 60,119)( 61,118)( 62,117)( 63,112)( 64,111)( 65,113)( 66,132)
( 67,134)( 68,133)( 69,137)( 70,136)( 71,135)( 72,130)( 73,129)( 74,131)
( 75,123)( 76,125)( 77,124)( 78,128)( 79,127)( 80,126)( 81,121)( 82,120)
( 83,122)(165,183)(166,185)(167,184)(168,188)(169,187)(170,186)(171,190)
(172,189)(173,191)(175,176)(177,179)(180,181)(192,244)(193,243)(194,245)
(195,237)(196,239)(197,238)(198,242)(199,241)(200,240)(201,235)(202,234)
(203,236)(204,228)(205,230)(206,229)(207,233)(208,232)(209,231)(210,226)
(211,225)(212,227)(213,219)(214,221)(215,220)(216,224)(217,223)(218,222);;
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*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(245)!(1,2);
s1 := Sym(245)!(  6, 10)(  7, 11)(  8,  9)( 15, 19)( 16, 20)( 17, 18)( 24, 28)
( 25, 29)( 26, 27)( 30, 57)( 31, 58)( 32, 59)( 33, 64)( 34, 65)( 35, 63)
( 36, 62)( 37, 60)( 38, 61)( 39, 66)( 40, 67)( 41, 68)( 42, 73)( 43, 74)
( 44, 72)( 45, 71)( 46, 69)( 47, 70)( 48, 75)( 49, 76)( 50, 77)( 51, 82)
( 52, 83)( 53, 81)( 54, 80)( 55, 78)( 56, 79)( 87, 91)( 88, 92)( 89, 90)
( 96,100)( 97,101)( 98, 99)(105,109)(106,110)(107,108)(111,138)(112,139)
(113,140)(114,145)(115,146)(116,144)(117,143)(118,141)(119,142)(120,147)
(121,148)(122,149)(123,154)(124,155)(125,153)(126,152)(127,150)(128,151)
(129,156)(130,157)(131,158)(132,163)(133,164)(134,162)(135,161)(136,159)
(137,160)(168,172)(169,173)(170,171)(177,181)(178,182)(179,180)(186,190)
(187,191)(188,189)(192,219)(193,220)(194,221)(195,226)(196,227)(197,225)
(198,224)(199,222)(200,223)(201,228)(202,229)(203,230)(204,235)(205,236)
(206,234)(207,233)(208,231)(209,232)(210,237)(211,238)(212,239)(213,244)
(214,245)(215,243)(216,242)(217,240)(218,241);
s2 := Sym(245)!(  3, 30)(  4, 32)(  5, 31)(  6, 33)(  7, 35)(  8, 34)(  9, 36)
( 10, 38)( 11, 37)( 12, 48)( 13, 50)( 14, 49)( 15, 51)( 16, 53)( 17, 52)
( 18, 54)( 19, 56)( 20, 55)( 21, 39)( 22, 41)( 23, 40)( 24, 42)( 25, 44)
( 26, 43)( 27, 45)( 28, 47)( 29, 46)( 58, 59)( 61, 62)( 64, 65)( 66, 75)
( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 73, 83)( 74, 82)
( 84,210)( 85,212)( 86,211)( 87,213)( 88,215)( 89,214)( 90,216)( 91,218)
( 92,217)( 93,201)( 94,203)( 95,202)( 96,204)( 97,206)( 98,205)( 99,207)
(100,209)(101,208)(102,192)(103,194)(104,193)(105,195)(106,197)(107,196)
(108,198)(109,200)(110,199)(111,183)(112,185)(113,184)(114,186)(115,188)
(116,187)(117,189)(118,191)(119,190)(120,174)(121,176)(122,175)(123,177)
(124,179)(125,178)(126,180)(127,182)(128,181)(129,165)(130,167)(131,166)
(132,168)(133,170)(134,169)(135,171)(136,173)(137,172)(138,237)(139,239)
(140,238)(141,240)(142,242)(143,241)(144,243)(145,245)(146,244)(147,228)
(148,230)(149,229)(150,231)(151,233)(152,232)(153,234)(154,236)(155,235)
(156,219)(157,221)(158,220)(159,222)(160,224)(161,223)(162,225)(163,227)
(164,226);
s3 := Sym(245)!(  3, 84)(  4, 86)(  5, 85)(  6, 89)(  7, 88)(  8, 87)(  9, 91)
( 10, 90)( 11, 92)( 12,102)( 13,104)( 14,103)( 15,107)( 16,106)( 17,105)
( 18,109)( 19,108)( 20,110)( 21, 93)( 22, 95)( 23, 94)( 24, 98)( 25, 97)
( 26, 96)( 27,100)( 28, 99)( 29,101)( 30,145)( 31,144)( 32,146)( 33,138)
( 34,140)( 35,139)( 36,143)( 37,142)( 38,141)( 39,163)( 40,162)( 41,164)
( 42,156)( 43,158)( 44,157)( 45,161)( 46,160)( 47,159)( 48,154)( 49,153)
( 50,155)( 51,147)( 52,149)( 53,148)( 54,152)( 55,151)( 56,150)( 57,114)
( 58,116)( 59,115)( 60,119)( 61,118)( 62,117)( 63,112)( 64,111)( 65,113)
( 66,132)( 67,134)( 68,133)( 69,137)( 70,136)( 71,135)( 72,130)( 73,129)
( 74,131)( 75,123)( 76,125)( 77,124)( 78,128)( 79,127)( 80,126)( 81,121)
( 82,120)( 83,122)(165,183)(166,185)(167,184)(168,188)(169,187)(170,186)
(171,190)(172,189)(173,191)(175,176)(177,179)(180,181)(192,244)(193,243)
(194,245)(195,237)(196,239)(197,238)(198,242)(199,241)(200,240)(201,235)
(202,234)(203,236)(204,228)(205,230)(206,229)(207,233)(208,232)(209,231)
(210,226)(211,225)(212,227)(213,219)(214,221)(215,220)(216,224)(217,223)
(218,222);
poly := sub<Sym(245)|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*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 >;

to this polytope