Polytope of Type {78,6}

Play with this polytope as a twisty puzzle

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

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