Polytope of Type {30,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {30,6}*1080c
if this polytope has a name.
Group : SmallGroup(1080,337)
Rank : 3
Schlafli Type : {30,6}
Number of vertices, edges, etc : 90, 270, 18
Order of s0s1s2 : 30
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {30,6}*540
   3-fold quotients : {30,6}*360b
   5-fold quotients : {6,6}*216b
   9-fold quotients : {10,6}*120, {30,2}*120
   10-fold quotients : {6,6}*108
   15-fold quotients : {6,6}*72a
   18-fold quotients : {15,2}*60
   27-fold quotients : {10,2}*40
   45-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {5,2}*20
   90-fold quotients : {2,3}*12, {3,2}*12
   135-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=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3.
      6 facets:
         6 of {30}*60
      50 vertex figures:
         20 of {6}*12
         30 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      10 facets:
         6 of {10}*20
         4 of {30}*60
      30 vertex figures:
         30 of {6}*12

Permutation Representation (GAP) : 
s0 := (  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)( 16, 31)( 17, 33)( 18, 32)( 19, 43)( 20, 45)( 21, 44)( 22, 40)( 23, 42)( 24, 41)( 25, 37)( 26, 39)( 27, 38)( 28, 34)( 29, 36)( 30, 35)( 47, 48)( 49, 58)( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 61, 76)( 62, 78)( 63, 77)( 64, 88)( 65, 90)( 66, 89)( 67, 85)( 68, 87)( 69, 86)( 70, 82)( 71, 84)( 72, 83)( 73, 79)( 74, 81)( 75, 80)( 92, 93)( 94,103)( 95,105)( 96,104)( 97,100)( 98,102)( 99,101)(106,121)(107,123)(108,122)(109,133)(110,135)(111,134)(112,130)(113,132)(114,131)(115,127)(116,129)(117,128)(118,124)(119,126)(120,125)(137,138)(139,148)(140,150)(141,149)(142,145)(143,147)(144,146)(151,166)(152,168)(153,167)(154,178)(155,180)(156,179)(157,175)(158,177)(159,176)(160,172)(161,174)(162,173)(163,169)(164,171)(165,170)(182,183)(184,193)(185,195)(186,194)(187,190)(188,192)(189,191)(196,211)(197,213)(198,212)(199,223)(200,225)(201,224)(202,220)(203,222)(204,221)(205,217)(206,219)(207,218)(208,214)(209,216)(210,215)(227,228)(229,238)(230,240)(231,239)(232,235)(233,237)(234,236)(241,256)(242,258)(243,257)(244,268)(245,270)(246,269)(247,265)(248,267)(249,266)(250,262)(251,264)(252,263)(253,259)(254,261)(255,260);;
s1 := (  1, 19)(  2, 20)(  3, 21)(  4, 16)(  5, 17)(  6, 18)(  7, 28)(  8, 29)(  9, 30)( 10, 25)( 11, 26)( 12, 27)( 13, 22)( 14, 23)( 15, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 43)( 38, 44)( 39, 45)( 46,109)( 47,110)( 48,111)( 49,106)( 50,107)( 51,108)( 52,118)( 53,119)( 54,120)( 55,115)( 56,116)( 57,117)( 58,112)( 59,113)( 60,114)( 61, 94)( 62, 95)( 63, 96)( 64, 91)( 65, 92)( 66, 93)( 67,103)( 68,104)( 69,105)( 70,100)( 71,101)( 72,102)( 73, 97)( 74, 98)( 75, 99)( 76,124)( 77,125)( 78,126)( 79,121)( 80,122)( 81,123)( 82,133)( 83,134)( 84,135)( 85,130)( 86,131)( 87,132)( 88,127)( 89,128)( 90,129)(136,154)(137,155)(138,156)(139,151)(140,152)(141,153)(142,163)(143,164)(144,165)(145,160)(146,161)(147,162)(148,157)(149,158)(150,159)(166,169)(167,170)(168,171)(172,178)(173,179)(174,180)(181,244)(182,245)(183,246)(184,241)(185,242)(186,243)(187,253)(188,254)(189,255)(190,250)(191,251)(192,252)(193,247)(194,248)(195,249)(196,229)(197,230)(198,231)(199,226)(200,227)(201,228)(202,238)(203,239)(204,240)(205,235)(206,236)(207,237)(208,232)(209,233)(210,234)(211,259)(212,260)(213,261)(214,256)(215,257)(216,258)(217,268)(218,269)(219,270)(220,265)(221,266)(222,267)(223,262)(224,263)(225,264);;
s2 := (  1,181)(  2,183)(  3,182)(  4,184)(  5,186)(  6,185)(  7,187)(  8,189)(  9,188)( 10,190)( 11,192)( 12,191)( 13,193)( 14,195)( 15,194)( 16,197)( 17,196)( 18,198)( 19,200)( 20,199)( 21,201)( 22,203)( 23,202)( 24,204)( 25,206)( 26,205)( 27,207)( 28,209)( 29,208)( 30,210)( 31,213)( 32,212)( 33,211)( 34,216)( 35,215)( 36,214)( 37,219)( 38,218)( 39,217)( 40,222)( 41,221)( 42,220)( 43,225)( 44,224)( 45,223)( 46,136)( 47,138)( 48,137)( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,152)( 62,151)( 63,153)( 64,155)( 65,154)( 66,156)( 67,158)( 68,157)( 69,159)( 70,161)( 71,160)( 72,162)( 73,164)( 74,163)( 75,165)( 76,168)( 77,167)( 78,166)( 79,171)( 80,170)( 81,169)( 82,174)( 83,173)( 84,172)( 85,177)( 86,176)( 87,175)( 88,180)( 89,179)( 90,178)( 91,226)( 92,228)( 93,227)( 94,229)( 95,231)( 96,230)( 97,232)( 98,234)( 99,233)(100,235)(101,237)(102,236)(103,238)(104,240)(105,239)(106,242)(107,241)(108,243)(109,245)(110,244)(111,246)(112,248)(113,247)(114,249)(115,251)(116,250)(117,252)(118,254)(119,253)(120,255)(121,258)(122,257)(123,256)(124,261)(125,260)(126,259)(127,264)(128,263)(129,262)(130,267)(131,266)(132,265)(133,270)(134,269)(135,268);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(270)!(  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)( 16, 31)( 17, 33)( 18, 32)( 19, 43)( 20, 45)( 21, 44)( 22, 40)( 23, 42)( 24, 41)( 25, 37)( 26, 39)( 27, 38)( 28, 34)( 29, 36)( 30, 35)( 47, 48)( 49, 58)( 50, 60)( 51, 59)( 52, 55)( 53, 57)( 54, 56)( 61, 76)( 62, 78)( 63, 77)( 64, 88)( 65, 90)( 66, 89)( 67, 85)( 68, 87)( 69, 86)( 70, 82)( 71, 84)( 72, 83)( 73, 79)( 74, 81)( 75, 80)( 92, 93)( 94,103)( 95,105)( 96,104)( 97,100)( 98,102)( 99,101)(106,121)(107,123)(108,122)(109,133)(110,135)(111,134)(112,130)(113,132)(114,131)(115,127)(116,129)(117,128)(118,124)(119,126)(120,125)(137,138)(139,148)(140,150)(141,149)(142,145)(143,147)(144,146)(151,166)(152,168)(153,167)(154,178)(155,180)(156,179)(157,175)(158,177)(159,176)(160,172)(161,174)(162,173)(163,169)(164,171)(165,170)(182,183)(184,193)(185,195)(186,194)(187,190)(188,192)(189,191)(196,211)(197,213)(198,212)(199,223)(200,225)(201,224)(202,220)(203,222)(204,221)(205,217)(206,219)(207,218)(208,214)(209,216)(210,215)(227,228)(229,238)(230,240)(231,239)(232,235)(233,237)(234,236)(241,256)(242,258)(243,257)(244,268)(245,270)(246,269)(247,265)(248,267)(249,266)(250,262)(251,264)(252,263)(253,259)(254,261)(255,260);
s1 := Sym(270)!(  1, 19)(  2, 20)(  3, 21)(  4, 16)(  5, 17)(  6, 18)(  7, 28)(  8, 29)(  9, 30)( 10, 25)( 11, 26)( 12, 27)( 13, 22)( 14, 23)( 15, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 43)( 38, 44)( 39, 45)( 46,109)( 47,110)( 48,111)( 49,106)( 50,107)( 51,108)( 52,118)( 53,119)( 54,120)( 55,115)( 56,116)( 57,117)( 58,112)( 59,113)( 60,114)( 61, 94)( 62, 95)( 63, 96)( 64, 91)( 65, 92)( 66, 93)( 67,103)( 68,104)( 69,105)( 70,100)( 71,101)( 72,102)( 73, 97)( 74, 98)( 75, 99)( 76,124)( 77,125)( 78,126)( 79,121)( 80,122)( 81,123)( 82,133)( 83,134)( 84,135)( 85,130)( 86,131)( 87,132)( 88,127)( 89,128)( 90,129)(136,154)(137,155)(138,156)(139,151)(140,152)(141,153)(142,163)(143,164)(144,165)(145,160)(146,161)(147,162)(148,157)(149,158)(150,159)(166,169)(167,170)(168,171)(172,178)(173,179)(174,180)(181,244)(182,245)(183,246)(184,241)(185,242)(186,243)(187,253)(188,254)(189,255)(190,250)(191,251)(192,252)(193,247)(194,248)(195,249)(196,229)(197,230)(198,231)(199,226)(200,227)(201,228)(202,238)(203,239)(204,240)(205,235)(206,236)(207,237)(208,232)(209,233)(210,234)(211,259)(212,260)(213,261)(214,256)(215,257)(216,258)(217,268)(218,269)(219,270)(220,265)(221,266)(222,267)(223,262)(224,263)(225,264);
s2 := Sym(270)!(  1,181)(  2,183)(  3,182)(  4,184)(  5,186)(  6,185)(  7,187)(  8,189)(  9,188)( 10,190)( 11,192)( 12,191)( 13,193)( 14,195)( 15,194)( 16,197)( 17,196)( 18,198)( 19,200)( 20,199)( 21,201)( 22,203)( 23,202)( 24,204)( 25,206)( 26,205)( 27,207)( 28,209)( 29,208)( 30,210)( 31,213)( 32,212)( 33,211)( 34,216)( 35,215)( 36,214)( 37,219)( 38,218)( 39,217)( 40,222)( 41,221)( 42,220)( 43,225)( 44,224)( 45,223)( 46,136)( 47,138)( 48,137)( 49,139)( 50,141)( 51,140)( 52,142)( 53,144)( 54,143)( 55,145)( 56,147)( 57,146)( 58,148)( 59,150)( 60,149)( 61,152)( 62,151)( 63,153)( 64,155)( 65,154)( 66,156)( 67,158)( 68,157)( 69,159)( 70,161)( 71,160)( 72,162)( 73,164)( 74,163)( 75,165)( 76,168)( 77,167)( 78,166)( 79,171)( 80,170)( 81,169)( 82,174)( 83,173)( 84,172)( 85,177)( 86,176)( 87,175)( 88,180)( 89,179)( 90,178)( 91,226)( 92,228)( 93,227)( 94,229)( 95,231)( 96,230)( 97,232)( 98,234)( 99,233)(100,235)(101,237)(102,236)(103,238)(104,240)(105,239)(106,242)(107,241)(108,243)(109,245)(110,244)(111,246)(112,248)(113,247)(114,249)(115,251)(116,250)(117,252)(118,254)(119,253)(120,255)(121,258)(122,257)(123,256)(124,261)(125,260)(126,259)(127,264)(128,263)(129,262)(130,267)(131,266)(132,265)(133,270)(134,269)(135,268);
poly := sub<Sym(270)|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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle