Polytope of Type {6,78}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,78}*936c
if this polytope has a name.
Group : SmallGroup(936,212)
Rank : 3
Schlafli Type : {6,78}
Number of vertices, edges, etc : 6, 234, 78
Order of s0s1s2 : 78
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,78,2} of size 1872
Vertex Figure Of :
   {2,6,78} of size 1872
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,39}*468
   3-fold quotients : {2,78}*312
   6-fold quotients : {2,39}*156
   9-fold quotients : {2,26}*104
   13-fold quotients : {6,6}*72b
   18-fold quotients : {2,13}*52
   26-fold quotients : {6,3}*36
   39-fold quotients : {2,6}*24
   78-fold quotients : {2,3}*12
   117-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,156}*1872c, {12,78}*1872c
Irregular Quotients (of which this is a minimal cover):
   None.

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