Polytope of Type {6,54}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,54}*1944g
if this polytope has a name.
Group : SmallGroup(1944,2343)
Rank : 3
Schlafli Type : {6,54}
Number of vertices, edges, etc : 18, 486, 162
Order of s₀s₁s₂ : 54
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,54}*648a, {6,54}*648b, {6,18}*648i
   6-fold quotients : {6,27}*324
   9-fold quotients : {2,54}*216, {6,18}*216a, {6,18}*216b, {6,6}*216d
   18-fold quotients : {2,27}*108, {6,9}*108
   27-fold quotients : {2,18}*72, {6,6}*72a, {6,6}*72b, {6,6}*72c
   54-fold quotients : {2,9}*36, {3,6}*36, {6,3}*36
   81-fold quotients : {2,6}*24, {6,2}*24
   162-fold quotients : {2,3}*12, {3,2}*12
   243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope