Polytope of Type {2,78,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,78,6}*1872b
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) :
   3-fold quotients : {2,26,6}*624, {2,78,2}*624
   6-fold quotients : {2,39,2}*312
   9-fold quotients : {2,26,2}*208
   13-fold quotients : {2,6,6}*144a
   18-fold quotients : {2,13,2}*104
   39-fold quotients : {2,2,6}*48, {2,6,2}*48
   78-fold quotients : {2,2,3}*24, {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)( 43, 54)( 44, 53)( 45, 52)( 46, 51)( 47, 50)
( 48, 49)( 55, 68)( 56, 80)( 57, 79)( 58, 78)( 59, 77)( 60, 76)( 61, 75)
( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 82, 93)( 83, 92)
( 84, 91)( 85, 90)( 86, 89)( 87, 88)( 94,107)( 95,119)( 96,118)( 97,117)
( 98,116)( 99,115)(100,114)(101,113)(102,112)(103,111)(104,110)(105,109)
(106,108)(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)(160,171)(161,170)(162,169)(163,168)
(164,167)(165,166)(172,185)(173,197)(174,196)(175,195)(176,194)(177,193)
(178,192)(179,191)(180,190)(181,189)(182,188)(183,187)(184,186)(199,210)
(200,209)(201,208)(202,207)(203,206)(204,205)(211,224)(212,236)(213,235)
(214,234)(215,233)(216,232)(217,231)(218,230)(219,229)(220,228)(221,227)
(222,226)(223,225);;
s2 := (  3, 17)(  4, 16)(  5, 28)(  6, 27)(  7, 26)(  8, 25)(  9, 24)( 10, 23)
( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 18)( 29, 30)( 31, 41)( 32, 40)
( 33, 39)( 34, 38)( 35, 37)( 42, 95)( 43, 94)( 44,106)( 45,105)( 46,104)
( 47,103)( 48,102)( 49,101)( 50,100)( 51, 99)( 52, 98)( 53, 97)( 54, 96)
( 55, 82)( 56, 81)( 57, 93)( 58, 92)( 59, 91)( 60, 90)( 61, 89)( 62, 88)
( 63, 87)( 64, 86)( 65, 85)( 66, 84)( 67, 83)( 68,108)( 69,107)( 70,119)
( 71,118)( 72,117)( 73,116)( 74,115)( 75,114)( 76,113)( 77,112)( 78,111)
( 79,110)( 80,109)(120,134)(121,133)(122,145)(123,144)(124,143)(125,142)
(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(146,147)
(148,158)(149,157)(150,156)(151,155)(152,154)(159,212)(160,211)(161,223)
(162,222)(163,221)(164,220)(165,219)(166,218)(167,217)(168,216)(169,215)
(170,214)(171,213)(172,199)(173,198)(174,210)(175,209)(176,208)(177,207)
(178,206)(179,205)(180,204)(181,203)(182,202)(183,201)(184,200)(185,225)
(186,224)(187,236)(188,235)(189,234)(190,233)(191,232)(192,231)(193,230)
(194,229)(195,228)(196,227)(197,226);;
s3 := (  3,159)(  4,160)(  5,161)(  6,162)(  7,163)(  8,164)(  9,165)( 10,166)
( 11,167)( 12,168)( 13,169)( 14,170)( 15,171)( 16,172)( 17,173)( 18,174)
( 19,175)( 20,176)( 21,177)( 22,178)( 23,179)( 24,180)( 25,181)( 26,182)
( 27,183)( 28,184)( 29,185)( 30,186)( 31,187)( 32,188)( 33,189)( 34,190)
( 35,191)( 36,192)( 37,193)( 38,194)( 39,195)( 40,196)( 41,197)( 42,120)
( 43,121)( 44,122)( 45,123)( 46,124)( 47,125)( 48,126)( 49,127)( 50,128)
( 51,129)( 52,130)( 53,131)( 54,132)( 55,133)( 56,134)( 57,135)( 58,136)
( 59,137)( 60,138)( 61,139)( 62,140)( 63,141)( 64,142)( 65,143)( 66,144)
( 67,145)( 68,146)( 69,147)( 70,148)( 71,149)( 72,150)( 73,151)( 74,152)
( 75,153)( 76,154)( 77,155)( 78,156)( 79,157)( 80,158)( 81,198)( 82,199)
( 83,200)( 84,201)( 85,202)( 86,203)( 87,204)( 88,205)( 89,206)( 90,207)
( 91,208)( 92,209)( 93,210)( 94,211)( 95,212)( 96,213)( 97,214)( 98,215)
( 99,216)(100,217)(101,218)(102,219)(103,220)(104,221)(105,222)(106,223)
(107,224)(108,225)(109,226)(110,227)(111,228)(112,229)(113,230)(114,231)
(115,232)(116,233)(117,234)(118,235)(119,236);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)( 43, 54)( 44, 53)( 45, 52)( 46, 51)
( 47, 50)( 48, 49)( 55, 68)( 56, 80)( 57, 79)( 58, 78)( 59, 77)( 60, 76)
( 61, 75)( 62, 74)( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 82, 93)
( 83, 92)( 84, 91)( 85, 90)( 86, 89)( 87, 88)( 94,107)( 95,119)( 96,118)
( 97,117)( 98,116)( 99,115)(100,114)(101,113)(102,112)(103,111)(104,110)
(105,109)(106,108)(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)(160,171)(161,170)(162,169)
(163,168)(164,167)(165,166)(172,185)(173,197)(174,196)(175,195)(176,194)
(177,193)(178,192)(179,191)(180,190)(181,189)(182,188)(183,187)(184,186)
(199,210)(200,209)(201,208)(202,207)(203,206)(204,205)(211,224)(212,236)
(213,235)(214,234)(215,233)(216,232)(217,231)(218,230)(219,229)(220,228)
(221,227)(222,226)(223,225);
s2 := Sym(236)!(  3, 17)(  4, 16)(  5, 28)(  6, 27)(  7, 26)(  8, 25)(  9, 24)
( 10, 23)( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 18)( 29, 30)( 31, 41)
( 32, 40)( 33, 39)( 34, 38)( 35, 37)( 42, 95)( 43, 94)( 44,106)( 45,105)
( 46,104)( 47,103)( 48,102)( 49,101)( 50,100)( 51, 99)( 52, 98)( 53, 97)
( 54, 96)( 55, 82)( 56, 81)( 57, 93)( 58, 92)( 59, 91)( 60, 90)( 61, 89)
( 62, 88)( 63, 87)( 64, 86)( 65, 85)( 66, 84)( 67, 83)( 68,108)( 69,107)
( 70,119)( 71,118)( 72,117)( 73,116)( 74,115)( 75,114)( 76,113)( 77,112)
( 78,111)( 79,110)( 80,109)(120,134)(121,133)(122,145)(123,144)(124,143)
(125,142)(126,141)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)
(146,147)(148,158)(149,157)(150,156)(151,155)(152,154)(159,212)(160,211)
(161,223)(162,222)(163,221)(164,220)(165,219)(166,218)(167,217)(168,216)
(169,215)(170,214)(171,213)(172,199)(173,198)(174,210)(175,209)(176,208)
(177,207)(178,206)(179,205)(180,204)(181,203)(182,202)(183,201)(184,200)
(185,225)(186,224)(187,236)(188,235)(189,234)(190,233)(191,232)(192,231)
(193,230)(194,229)(195,228)(196,227)(197,226);
s3 := Sym(236)!(  3,159)(  4,160)(  5,161)(  6,162)(  7,163)(  8,164)(  9,165)
( 10,166)( 11,167)( 12,168)( 13,169)( 14,170)( 15,171)( 16,172)( 17,173)
( 18,174)( 19,175)( 20,176)( 21,177)( 22,178)( 23,179)( 24,180)( 25,181)
( 26,182)( 27,183)( 28,184)( 29,185)( 30,186)( 31,187)( 32,188)( 33,189)
( 34,190)( 35,191)( 36,192)( 37,193)( 38,194)( 39,195)( 40,196)( 41,197)
( 42,120)( 43,121)( 44,122)( 45,123)( 46,124)( 47,125)( 48,126)( 49,127)
( 50,128)( 51,129)( 52,130)( 53,131)( 54,132)( 55,133)( 56,134)( 57,135)
( 58,136)( 59,137)( 60,138)( 61,139)( 62,140)( 63,141)( 64,142)( 65,143)
( 66,144)( 67,145)( 68,146)( 69,147)( 70,148)( 71,149)( 72,150)( 73,151)
( 74,152)( 75,153)( 76,154)( 77,155)( 78,156)( 79,157)( 80,158)( 81,198)
( 82,199)( 83,200)( 84,201)( 85,202)( 86,203)( 87,204)( 88,205)( 89,206)
( 90,207)( 91,208)( 92,209)( 93,210)( 94,211)( 95,212)( 96,213)( 97,214)
( 98,215)( 99,216)(100,217)(101,218)(102,219)(103,220)(104,221)(105,222)
(106,223)(107,224)(108,225)(109,226)(110,227)(111,228)(112,229)(113,230)
(114,231)(115,232)(116,233)(117,234)(118,235)(119,236);
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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