Polytope of Type {4,54}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,54}*432b
if this polytope has a name.
Group : SmallGroup(432,224)
Rank : 3
Schlafli Type : {4,54}
Number of vertices, edges, etc : 4, 108, 54
Order of s0s1s2 : 54
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,54,2} of size 864
   {4,54,4} of size 1728
   {4,54,4} of size 1728
   {4,54,4} of size 1728
Vertex Figure Of :
   {2,4,54} of size 864
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,27}*216
   3-fold quotients : {4,18}*144b
   6-fold quotients : {4,9}*72
   9-fold quotients : {4,6}*48c
   18-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,108}*864b, {4,108}*864c, {4,54}*864
   3-fold covers : {4,162}*1296b
   4-fold covers : {4,54}*1728a, {4,216}*1728c, {4,216}*1728d, {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c
Irregular Quotients (of which this is a minimal cover):
   None.

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

Twisty Puzzle