Polytope of Type {2,4,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,6,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,30882)
Rank : 5
Schlafli Type : {2,4,6,6}
Number of vertices, edges, etc : 2, 4, 36, 54, 18
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 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 : {2,4,6,3}*864a, {2,2,6,6}*864a
   3-fold quotients : {2,4,6,6}*576c
   4-fold quotients : {2,2,6,3}*432
   6-fold quotients : {2,4,6,3}*288, {2,2,6,6}*288b
   9-fold quotients : {2,4,2,6}*192
   12-fold quotients : {2,2,6,3}*144
   18-fold quotients : {2,4,2,3}*96, {2,2,2,6}*96
   27-fold quotients : {2,4,2,2}*64
   36-fold quotients : {2,2,2,3}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := (1,2);;
s1 := (  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)(  9, 63)( 10, 64)
( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)
( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)
( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)
( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)
( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)
( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110)(111,165)(112,166)
(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)
(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)
(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)
(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)
(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)
(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)
(161,215)(162,216)(163,217)(164,218);;
s2 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)( 64, 89)
( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)( 72, 96)
( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)( 80,109)
( 81,105)( 82,107)( 83,106)(112,113)(114,117)(115,119)(116,118)(121,122)
(123,126)(124,128)(125,127)(130,131)(132,135)(133,137)(134,136)(139,140)
(141,144)(142,146)(143,145)(148,149)(150,153)(151,155)(152,154)(157,158)
(159,162)(160,164)(161,163)(165,192)(166,194)(167,193)(168,198)(169,200)
(170,199)(171,195)(172,197)(173,196)(174,201)(175,203)(176,202)(177,207)
(178,209)(179,208)(180,204)(181,206)(182,205)(183,210)(184,212)(185,211)
(186,216)(187,218)(188,217)(189,213)(190,215)(191,214);;
s3 := (  3,  6)(  4,  7)(  5,  8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)( 16, 22)
( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)( 39, 51)
( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)( 47, 56)
( 57, 60)( 58, 61)( 59, 62)( 66, 78)( 67, 79)( 68, 80)( 69, 75)( 70, 76)
( 71, 77)( 72, 81)( 73, 82)( 74, 83)( 84, 87)( 85, 88)( 86, 89)( 93,105)
( 94,106)( 95,107)( 96,102)( 97,103)( 98,104)( 99,108)(100,109)(101,110)
(111,114)(112,115)(113,116)(120,132)(121,133)(122,134)(123,129)(124,130)
(125,131)(126,135)(127,136)(128,137)(138,141)(139,142)(140,143)(147,159)
(148,160)(149,161)(150,156)(151,157)(152,158)(153,162)(154,163)(155,164)
(165,168)(166,169)(167,170)(174,186)(175,187)(176,188)(177,183)(178,184)
(179,185)(180,189)(181,190)(182,191)(192,195)(193,196)(194,197)(201,213)
(202,214)(203,215)(204,210)(205,211)(206,212)(207,216)(208,217)(209,218);;
s4 := (  3,120)(  4,121)(  5,122)(  6,128)(  7,126)(  8,127)(  9,124)( 10,125)
( 11,123)( 12,111)( 13,112)( 14,113)( 15,119)( 16,117)( 17,118)( 18,115)
( 19,116)( 20,114)( 21,129)( 22,130)( 23,131)( 24,137)( 25,135)( 26,136)
( 27,133)( 28,134)( 29,132)( 30,147)( 31,148)( 32,149)( 33,155)( 34,153)
( 35,154)( 36,151)( 37,152)( 38,150)( 39,138)( 40,139)( 41,140)( 42,146)
( 43,144)( 44,145)( 45,142)( 46,143)( 47,141)( 48,156)( 49,157)( 50,158)
( 51,164)( 52,162)( 53,163)( 54,160)( 55,161)( 56,159)( 57,174)( 58,175)
( 59,176)( 60,182)( 61,180)( 62,181)( 63,178)( 64,179)( 65,177)( 66,165)
( 67,166)( 68,167)( 69,173)( 70,171)( 71,172)( 72,169)( 73,170)( 74,168)
( 75,183)( 76,184)( 77,185)( 78,191)( 79,189)( 80,190)( 81,187)( 82,188)
( 83,186)( 84,201)( 85,202)( 86,203)( 87,209)( 88,207)( 89,208)( 90,205)
( 91,206)( 92,204)( 93,192)( 94,193)( 95,194)( 96,200)( 97,198)( 98,199)
( 99,196)(100,197)(101,195)(102,210)(103,211)(104,212)(105,218)(106,216)
(107,217)(108,214)(109,215)(110,213);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3*s4*s2*s3*s2*s4*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)(  9, 63)
( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)
( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)
( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)
( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)
( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)
( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)( 55,109)( 56,110)(111,165)
(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)
(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)
(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)
(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)
(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)
(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)
(160,214)(161,215)(162,216)(163,217)(164,218);
s2 := Sym(218)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 57, 84)( 58, 86)( 59, 85)( 60, 90)( 61, 92)( 62, 91)( 63, 87)
( 64, 89)( 65, 88)( 66, 93)( 67, 95)( 68, 94)( 69, 99)( 70,101)( 71,100)
( 72, 96)( 73, 98)( 74, 97)( 75,102)( 76,104)( 77,103)( 78,108)( 79,110)
( 80,109)( 81,105)( 82,107)( 83,106)(112,113)(114,117)(115,119)(116,118)
(121,122)(123,126)(124,128)(125,127)(130,131)(132,135)(133,137)(134,136)
(139,140)(141,144)(142,146)(143,145)(148,149)(150,153)(151,155)(152,154)
(157,158)(159,162)(160,164)(161,163)(165,192)(166,194)(167,193)(168,198)
(169,200)(170,199)(171,195)(172,197)(173,196)(174,201)(175,203)(176,202)
(177,207)(178,209)(179,208)(180,204)(181,206)(182,205)(183,210)(184,212)
(185,211)(186,216)(187,218)(188,217)(189,213)(190,215)(191,214);
s3 := Sym(218)!(  3,  6)(  4,  7)(  5,  8)( 12, 24)( 13, 25)( 14, 26)( 15, 21)
( 16, 22)( 17, 23)( 18, 27)( 19, 28)( 20, 29)( 30, 33)( 31, 34)( 32, 35)
( 39, 51)( 40, 52)( 41, 53)( 42, 48)( 43, 49)( 44, 50)( 45, 54)( 46, 55)
( 47, 56)( 57, 60)( 58, 61)( 59, 62)( 66, 78)( 67, 79)( 68, 80)( 69, 75)
( 70, 76)( 71, 77)( 72, 81)( 73, 82)( 74, 83)( 84, 87)( 85, 88)( 86, 89)
( 93,105)( 94,106)( 95,107)( 96,102)( 97,103)( 98,104)( 99,108)(100,109)
(101,110)(111,114)(112,115)(113,116)(120,132)(121,133)(122,134)(123,129)
(124,130)(125,131)(126,135)(127,136)(128,137)(138,141)(139,142)(140,143)
(147,159)(148,160)(149,161)(150,156)(151,157)(152,158)(153,162)(154,163)
(155,164)(165,168)(166,169)(167,170)(174,186)(175,187)(176,188)(177,183)
(178,184)(179,185)(180,189)(181,190)(182,191)(192,195)(193,196)(194,197)
(201,213)(202,214)(203,215)(204,210)(205,211)(206,212)(207,216)(208,217)
(209,218);
s4 := Sym(218)!(  3,120)(  4,121)(  5,122)(  6,128)(  7,126)(  8,127)(  9,124)
( 10,125)( 11,123)( 12,111)( 13,112)( 14,113)( 15,119)( 16,117)( 17,118)
( 18,115)( 19,116)( 20,114)( 21,129)( 22,130)( 23,131)( 24,137)( 25,135)
( 26,136)( 27,133)( 28,134)( 29,132)( 30,147)( 31,148)( 32,149)( 33,155)
( 34,153)( 35,154)( 36,151)( 37,152)( 38,150)( 39,138)( 40,139)( 41,140)
( 42,146)( 43,144)( 44,145)( 45,142)( 46,143)( 47,141)( 48,156)( 49,157)
( 50,158)( 51,164)( 52,162)( 53,163)( 54,160)( 55,161)( 56,159)( 57,174)
( 58,175)( 59,176)( 60,182)( 61,180)( 62,181)( 63,178)( 64,179)( 65,177)
( 66,165)( 67,166)( 68,167)( 69,173)( 70,171)( 71,172)( 72,169)( 73,170)
( 74,168)( 75,183)( 76,184)( 77,185)( 78,191)( 79,189)( 80,190)( 81,187)
( 82,188)( 83,186)( 84,201)( 85,202)( 86,203)( 87,209)( 88,207)( 89,208)
( 90,205)( 91,206)( 92,204)( 93,192)( 94,193)( 95,194)( 96,200)( 97,198)
( 98,199)( 99,196)(100,197)(101,195)(102,210)(103,211)(104,212)(105,218)
(106,216)(107,217)(108,214)(109,215)(110,213);
poly := sub<Sym(218)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3*s4*s2*s3*s2*s4*s3 >;

to this polytope