Polytope of Type {2,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,6}*1944c
if this polytope has a name.
Group : SmallGroup(1944,949)
Rank : 4
Schlafli Type : {2,18,6}
Number of vertices, edges, etc : 2, 81, 243, 27
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {2,18,6}*648a, {2,6,6}*648b
   9-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope