Polytope of Type {18,6,2}

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

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

Permutation Representation (Magma) :

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

to this polytope