Polytope of Type {6,52}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,52}*1872
if this polytope has a name.
Group : SmallGroup(1872,1058)
Rank : 3
Schlafli Type : {6,52}
Number of vertices, edges, etc : 18, 468, 156
Order of s0s1s2 : 52
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) :
   2-fold quotients : {6,52}*936
   9-fold quotients : {2,52}*208
   13-fold quotients : {6,4}*144
   18-fold quotients : {2,26}*104
   26-fold quotients : {6,4}*72
   36-fold quotients : {2,13}*52
   117-fold quotients : {2,4}*16
   234-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s2*s1*s0*s1*s2> of order 3.
      104 facets:
         26 of {6}*12
         78 of {2}*4
      6 vertex figures:
         6 of {52}*104
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      52 facets:
         52 of {6}*12
      6 vertex figures:
         6 of {52}*104

Permutation Representation (GAP) : 
s0 := (  1,118)(  2,119)(  3,120)(  4,121)(  5,122)(  6,123)(  7,124)(  8,125)(  9,126)( 10,127)( 11,128)( 12,129)( 13,130)( 14,144)( 15,145)( 16,146)( 17,147)( 18,148)( 19,149)( 20,150)( 21,151)( 22,152)( 23,153)( 24,154)( 25,155)( 26,156)( 27,131)( 28,132)( 29,133)( 30,134)( 31,135)( 32,136)( 33,137)( 34,138)( 35,139)( 36,140)( 37,141)( 38,142)( 39,143)( 40,196)( 41,197)( 42,198)( 43,199)( 44,200)( 45,201)( 46,202)( 47,203)( 48,204)( 49,205)( 50,206)( 51,207)( 52,208)( 53,222)( 54,223)( 55,224)( 56,225)( 57,226)( 58,227)( 59,228)( 60,229)( 61,230)( 62,231)( 63,232)( 64,233)( 65,234)( 66,209)( 67,210)( 68,211)( 69,212)( 70,213)( 71,214)( 72,215)( 73,216)( 74,217)( 75,218)( 76,219)( 77,220)( 78,221)( 79,157)( 80,158)( 81,159)( 82,160)( 83,161)( 84,162)( 85,163)( 86,164)( 87,165)( 88,166)( 89,167)( 90,168)( 91,169)( 92,183)( 93,184)( 94,185)( 95,186)( 96,187)( 97,188)( 98,189)( 99,190)(100,191)(101,192)(102,193)(103,194)(104,195)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,181)(117,182);;
s1 := (  1, 40)(  2, 52)(  3, 51)(  4, 50)(  5, 49)(  6, 48)(  7, 47)(  8, 46)(  9, 45)( 10, 44)( 11, 43)( 12, 42)( 13, 41)( 14, 53)( 15, 65)( 16, 64)( 17, 63)( 18, 62)( 19, 61)( 20, 60)( 21, 59)( 22, 58)( 23, 57)( 24, 56)( 25, 55)( 26, 54)( 27, 66)( 28, 78)( 29, 77)( 30, 76)( 31, 75)( 32, 74)( 33, 73)( 34, 72)( 35, 71)( 36, 70)( 37, 69)( 38, 68)( 39, 67)( 80, 91)( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 93,104)( 94,103)( 95,102)( 96,101)( 97,100)( 98, 99)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(118,157)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,170)(132,182)(133,181)(134,180)(135,179)(136,178)(137,177)(138,176)(139,175)(140,174)(141,173)(142,172)(143,171)(144,183)(145,195)(146,194)(147,193)(148,192)(149,191)(150,190)(151,189)(152,188)(153,187)(154,186)(155,185)(156,184)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(210,221)(211,220)(212,219)(213,218)(214,217)(215,216)(223,234)(224,233)(225,232)(226,231)(227,230)(228,229);;
s2 := (  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 80)( 15, 79)( 16, 91)( 17, 90)( 18, 89)( 19, 88)( 20, 87)( 21, 86)( 22, 85)( 23, 84)( 24, 83)( 25, 82)( 26, 81)( 27, 41)( 28, 40)( 29, 52)( 30, 51)( 31, 50)( 32, 49)( 33, 48)( 34, 47)( 35, 46)( 36, 45)( 37, 44)( 38, 43)( 39, 42)( 53,106)( 54,105)( 55,117)( 56,116)( 57,115)( 58,114)( 59,113)( 60,112)( 61,111)( 62,110)( 63,109)( 64,108)( 65,107)( 66, 67)( 68, 78)( 69, 77)( 70, 76)( 71, 75)( 72, 74)( 92, 93)( 94,104)( 95,103)( 96,102)( 97,101)( 98,100)(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)(131,197)(132,196)(133,208)(134,207)(135,206)(136,205)(137,204)(138,203)(139,202)(140,201)(141,200)(142,199)(143,198)(144,158)(145,157)(146,169)(147,168)(148,167)(149,166)(150,165)(151,164)(152,163)(153,162)(154,161)(155,160)(156,159)(170,223)(171,222)(172,234)(173,233)(174,232)(175,231)(176,230)(177,229)(178,228)(179,227)(180,226)(181,225)(182,224)(183,184)(185,195)(186,194)(187,193)(188,192)(189,191)(209,210)(211,221)(212,220)(213,219)(214,218)(215,217);;
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, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(234)!(  1,118)(  2,119)(  3,120)(  4,121)(  5,122)(  6,123)(  7,124)(  8,125)(  9,126)( 10,127)( 11,128)( 12,129)( 13,130)( 14,144)( 15,145)( 16,146)( 17,147)( 18,148)( 19,149)( 20,150)( 21,151)( 22,152)( 23,153)( 24,154)( 25,155)( 26,156)( 27,131)( 28,132)( 29,133)( 30,134)( 31,135)( 32,136)( 33,137)( 34,138)( 35,139)( 36,140)( 37,141)( 38,142)( 39,143)( 40,196)( 41,197)( 42,198)( 43,199)( 44,200)( 45,201)( 46,202)( 47,203)( 48,204)( 49,205)( 50,206)( 51,207)( 52,208)( 53,222)( 54,223)( 55,224)( 56,225)( 57,226)( 58,227)( 59,228)( 60,229)( 61,230)( 62,231)( 63,232)( 64,233)( 65,234)( 66,209)( 67,210)( 68,211)( 69,212)( 70,213)( 71,214)( 72,215)( 73,216)( 74,217)( 75,218)( 76,219)( 77,220)( 78,221)( 79,157)( 80,158)( 81,159)( 82,160)( 83,161)( 84,162)( 85,163)( 86,164)( 87,165)( 88,166)( 89,167)( 90,168)( 91,169)( 92,183)( 93,184)( 94,185)( 95,186)( 96,187)( 97,188)( 98,189)( 99,190)(100,191)(101,192)(102,193)(103,194)(104,195)(105,170)(106,171)(107,172)(108,173)(109,174)(110,175)(111,176)(112,177)(113,178)(114,179)(115,180)(116,181)(117,182);
s1 := Sym(234)!(  1, 40)(  2, 52)(  3, 51)(  4, 50)(  5, 49)(  6, 48)(  7, 47)(  8, 46)(  9, 45)( 10, 44)( 11, 43)( 12, 42)( 13, 41)( 14, 53)( 15, 65)( 16, 64)( 17, 63)( 18, 62)( 19, 61)( 20, 60)( 21, 59)( 22, 58)( 23, 57)( 24, 56)( 25, 55)( 26, 54)( 27, 66)( 28, 78)( 29, 77)( 30, 76)( 31, 75)( 32, 74)( 33, 73)( 34, 72)( 35, 71)( 36, 70)( 37, 69)( 38, 68)( 39, 67)( 80, 91)( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 93,104)( 94,103)( 95,102)( 96,101)( 97,100)( 98, 99)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(118,157)(119,169)(120,168)(121,167)(122,166)(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,170)(132,182)(133,181)(134,180)(135,179)(136,178)(137,177)(138,176)(139,175)(140,174)(141,173)(142,172)(143,171)(144,183)(145,195)(146,194)(147,193)(148,192)(149,191)(150,190)(151,189)(152,188)(153,187)(154,186)(155,185)(156,184)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(210,221)(211,220)(212,219)(213,218)(214,217)(215,216)(223,234)(224,233)(225,232)(226,231)(227,230)(228,229);
s2 := Sym(234)!(  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 80)( 15, 79)( 16, 91)( 17, 90)( 18, 89)( 19, 88)( 20, 87)( 21, 86)( 22, 85)( 23, 84)( 24, 83)( 25, 82)( 26, 81)( 27, 41)( 28, 40)( 29, 52)( 30, 51)( 31, 50)( 32, 49)( 33, 48)( 34, 47)( 35, 46)( 36, 45)( 37, 44)( 38, 43)( 39, 42)( 53,106)( 54,105)( 55,117)( 56,116)( 57,115)( 58,114)( 59,113)( 60,112)( 61,111)( 62,110)( 63,109)( 64,108)( 65,107)( 66, 67)( 68, 78)( 69, 77)( 70, 76)( 71, 75)( 72, 74)( 92, 93)( 94,104)( 95,103)( 96,102)( 97,101)( 98,100)(118,119)(120,130)(121,129)(122,128)(123,127)(124,126)(131,197)(132,196)(133,208)(134,207)(135,206)(136,205)(137,204)(138,203)(139,202)(140,201)(141,200)(142,199)(143,198)(144,158)(145,157)(146,169)(147,168)(148,167)(149,166)(150,165)(151,164)(152,163)(153,162)(154,161)(155,160)(156,159)(170,223)(171,222)(172,234)(173,233)(174,232)(175,231)(176,230)(177,229)(178,228)(179,227)(180,226)(181,225)(182,224)(183,184)(185,195)(186,194)(187,193)(188,192)(189,191)(209,210)(211,221)(212,220)(213,219)(214,218)(215,217);
poly := sub<Sym(234)|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, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*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 >;
References : None.
to this polytope

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