Polytope of Type {2,78,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,78,6}*1872c
if this polytope has a name.
Group : SmallGroup(1872,1084)
Rank : 4
Schlafli Type : {2,78,6}
Number of vertices, edges, etc : 2, 78, 234, 6
Order of s₀s₁s₂s₃ : 78
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
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 : {2,39,6}*936
   3-fold quotients : {2,78,2}*624
   6-fold quotients : {2,39,2}*312
   9-fold quotients : {2,26,2}*208
   13-fold quotients : {2,6,6}*144c
   18-fold quotients : {2,13,2}*104
   26-fold quotients : {2,3,6}*72
   39-fold quotients : {2,6,2}*48
   78-fold quotients : {2,3,2}*24
   117-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)( 16, 29)( 17, 41)
( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 22, 36)( 23, 35)( 24, 34)( 25, 33)
( 26, 32)( 27, 31)( 28, 30)( 42, 81)( 43, 93)( 44, 92)( 45, 91)( 46, 90)
( 47, 89)( 48, 88)( 49, 87)( 50, 86)( 51, 85)( 52, 84)( 53, 83)( 54, 82)
( 55,107)( 56,119)( 57,118)( 58,117)( 59,116)( 60,115)( 61,114)( 62,113)
( 63,112)( 64,111)( 65,110)( 66,109)( 67,108)( 68, 94)( 69,106)( 70,105)
( 71,104)( 72,103)( 73,102)( 74,101)( 75,100)( 76, 99)( 77, 98)( 78, 97)
( 79, 96)( 80, 95)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)
(133,146)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)(140,152)
(141,151)(142,150)(143,149)(144,148)(145,147)(159,198)(160,210)(161,209)
(162,208)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)
(170,200)(171,199)(172,224)(173,236)(174,235)(175,234)(176,233)(177,232)
(178,231)(179,230)(180,229)(181,228)(182,227)(183,226)(184,225)(185,211)
(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)(193,216)
(194,215)(195,214)(196,213)(197,212);;
s2 := (  3,173)(  4,172)(  5,184)(  6,183)(  7,182)(  8,181)(  9,180)( 10,179)
( 11,178)( 12,177)( 13,176)( 14,175)( 15,174)( 16,160)( 17,159)( 18,171)
( 19,170)( 20,169)( 21,168)( 22,167)( 23,166)( 24,165)( 25,164)( 26,163)
( 27,162)( 28,161)( 29,186)( 30,185)( 31,197)( 32,196)( 33,195)( 34,194)
( 35,193)( 36,192)( 37,191)( 38,190)( 39,189)( 40,188)( 41,187)( 42,134)
( 43,133)( 44,145)( 45,144)( 46,143)( 47,142)( 48,141)( 49,140)( 50,139)
( 51,138)( 52,137)( 53,136)( 54,135)( 55,121)( 56,120)( 57,132)( 58,131)
( 59,130)( 60,129)( 61,128)( 62,127)( 63,126)( 64,125)( 65,124)( 66,123)
( 67,122)( 68,147)( 69,146)( 70,158)( 71,157)( 72,156)( 73,155)( 74,154)
( 75,153)( 76,152)( 77,151)( 78,150)( 79,149)( 80,148)( 81,212)( 82,211)
( 83,223)( 84,222)( 85,221)( 86,220)( 87,219)( 88,218)( 89,217)( 90,216)
( 91,215)( 92,214)( 93,213)( 94,199)( 95,198)( 96,210)( 97,209)( 98,208)
( 99,207)(100,206)(101,205)(102,204)(103,203)(104,202)(105,201)(106,200)
(107,225)(108,224)(109,236)(110,235)(111,234)(112,233)(113,232)(114,231)
(115,230)(116,229)(117,228)(118,227)(119,226);;
s3 := ( 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)( 79,118)( 80,119)(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)(196,235)(197,236);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(236)!(1,2);
s1 := Sym(236)!(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)( 16, 29)
( 17, 41)( 18, 40)( 19, 39)( 20, 38)( 21, 37)( 22, 36)( 23, 35)( 24, 34)
( 25, 33)( 26, 32)( 27, 31)( 28, 30)( 42, 81)( 43, 93)( 44, 92)( 45, 91)
( 46, 90)( 47, 89)( 48, 88)( 49, 87)( 50, 86)( 51, 85)( 52, 84)( 53, 83)
( 54, 82)( 55,107)( 56,119)( 57,118)( 58,117)( 59,116)( 60,115)( 61,114)
( 62,113)( 63,112)( 64,111)( 65,110)( 66,109)( 67,108)( 68, 94)( 69,106)
( 70,105)( 71,104)( 72,103)( 73,102)( 74,101)( 75,100)( 76, 99)( 77, 98)
( 78, 97)( 79, 96)( 80, 95)(121,132)(122,131)(123,130)(124,129)(125,128)
(126,127)(133,146)(134,158)(135,157)(136,156)(137,155)(138,154)(139,153)
(140,152)(141,151)(142,150)(143,149)(144,148)(145,147)(159,198)(160,210)
(161,209)(162,208)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)
(169,201)(170,200)(171,199)(172,224)(173,236)(174,235)(175,234)(176,233)
(177,232)(178,231)(179,230)(180,229)(181,228)(182,227)(183,226)(184,225)
(185,211)(186,223)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)
(193,216)(194,215)(195,214)(196,213)(197,212);
s2 := Sym(236)!(  3,173)(  4,172)(  5,184)(  6,183)(  7,182)(  8,181)(  9,180)
( 10,179)( 11,178)( 12,177)( 13,176)( 14,175)( 15,174)( 16,160)( 17,159)
( 18,171)( 19,170)( 20,169)( 21,168)( 22,167)( 23,166)( 24,165)( 25,164)
( 26,163)( 27,162)( 28,161)( 29,186)( 30,185)( 31,197)( 32,196)( 33,195)
( 34,194)( 35,193)( 36,192)( 37,191)( 38,190)( 39,189)( 40,188)( 41,187)
( 42,134)( 43,133)( 44,145)( 45,144)( 46,143)( 47,142)( 48,141)( 49,140)
( 50,139)( 51,138)( 52,137)( 53,136)( 54,135)( 55,121)( 56,120)( 57,132)
( 58,131)( 59,130)( 60,129)( 61,128)( 62,127)( 63,126)( 64,125)( 65,124)
( 66,123)( 67,122)( 68,147)( 69,146)( 70,158)( 71,157)( 72,156)( 73,155)
( 74,154)( 75,153)( 76,152)( 77,151)( 78,150)( 79,149)( 80,148)( 81,212)
( 82,211)( 83,223)( 84,222)( 85,221)( 86,220)( 87,219)( 88,218)( 89,217)
( 90,216)( 91,215)( 92,214)( 93,213)( 94,199)( 95,198)( 96,210)( 97,209)
( 98,208)( 99,207)(100,206)(101,205)(102,204)(103,203)(104,202)(105,201)
(106,200)(107,225)(108,224)(109,236)(110,235)(111,234)(112,233)(113,232)
(114,231)(115,230)(116,229)(117,228)(118,227)(119,226);
s3 := Sym(236)!( 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)( 79,118)( 80,119)
(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)(196,235)(197,236);
poly := sub<Sym(236)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope