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}*1728g
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 5
Schlafli Type : {2,4,6,6}
Number of vertices, edges, etc : 2, 8, 36, 54, 9
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,4,6,6}*864
4-fold quotients : {2,2,6,6}*432
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,113)( 4,114)( 5,111)( 6,112)( 7,117)( 8,118)( 9,115)( 10,116)( 11,121)( 12,122)( 13,119)( 14,120)( 15,125)( 16,126)( 17,123)( 18,124)( 19,129)( 20,130)( 21,127)( 22,128)( 23,133)( 24,134)( 25,131)( 26,132)( 27,137)( 28,138)( 29,135)( 30,136)( 31,141)( 32,142)( 33,139)( 34,140)( 35,145)( 36,146)( 37,143)( 38,144)( 39,149)( 40,150)( 41,147)( 42,148)( 43,153)( 44,154)( 45,151)( 46,152)( 47,157)( 48,158)( 49,155)( 50,156)( 51,161)( 52,162)( 53,159)( 54,160)( 55,165)( 56,166)( 57,163)( 58,164)( 59,169)( 60,170)( 61,167)( 62,168)( 63,173)( 64,174)( 65,171)( 66,172)( 67,177)( 68,178)( 69,175)( 70,176)( 71,181)( 72,182)( 73,179)( 74,180)( 75,185)( 76,186)( 77,183)( 78,184)( 79,189)( 80,190)( 81,187)( 82,188)( 83,193)( 84,194)( 85,191)( 86,192)( 87,197)( 88,198)( 89,195)( 90,196)( 91,201)( 92,202)( 93,199)( 94,200)( 95,205)( 96,206)( 97,203)( 98,204)( 99,209)(100,210)(101,207)(102,208)(103,213)(104,214)(105,211)(106,212)(107,217)(108,218)(109,215)(110,216);;
s2 := ( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)( 47, 79)( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)( 63, 99)( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)( 71,103)( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)(118,122)(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)(140,145)(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)(152,193)(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)(160,197)(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)(168,201)(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)(176,217)(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);;
s3 := ( 3, 75)( 4, 78)( 5, 77)( 6, 76)( 7, 79)( 8, 82)( 9, 81)( 10, 80)( 11, 83)( 12, 86)( 13, 85)( 14, 84)( 15,103)( 16,106)( 17,105)( 18,104)( 19,107)( 20,110)( 21,109)( 22,108)( 23, 99)( 24,102)( 25,101)( 26,100)( 27, 95)( 28, 98)( 29, 97)( 30, 96)( 31, 87)( 32, 90)( 33, 89)( 34, 88)( 35, 91)( 36, 94)( 37, 93)( 38, 92)( 40, 42)( 44, 46)( 48, 50)( 51, 67)( 52, 70)( 53, 69)( 54, 68)( 55, 71)( 56, 74)( 57, 73)( 58, 72)( 59, 63)( 60, 66)( 61, 65)( 62, 64)(111,183)(112,186)(113,185)(114,184)(115,187)(116,190)(117,189)(118,188)(119,191)(120,194)(121,193)(122,192)(123,211)(124,214)(125,213)(126,212)(127,215)(128,218)(129,217)(130,216)(131,207)(132,210)(133,209)(134,208)(135,203)(136,206)(137,205)(138,204)(139,195)(140,198)(141,197)(142,196)(143,199)(144,202)(145,201)(146,200)(148,150)(152,154)(156,158)(159,175)(160,178)(161,177)(162,176)(163,179)(164,182)(165,181)(166,180)(167,171)(168,174)(169,173)(170,172);;
s4 := ( 3, 15)( 4, 16)( 5, 17)( 6, 18)( 7, 23)( 8, 24)( 9, 25)( 10, 26)( 11, 19)( 12, 20)( 13, 21)( 14, 22)( 31, 35)( 32, 36)( 33, 37)( 34, 38)( 39, 51)( 40, 52)( 41, 53)( 42, 54)( 43, 59)( 44, 60)( 45, 61)( 46, 62)( 47, 55)( 48, 56)( 49, 57)( 50, 58)( 67, 71)( 68, 72)( 69, 73)( 70, 74)( 75, 87)( 76, 88)( 77, 89)( 78, 90)( 79, 95)( 80, 96)( 81, 97)( 82, 98)( 83, 91)( 84, 92)( 85, 93)( 86, 94)(103,107)(104,108)(105,109)(106,110)(111,123)(112,124)(113,125)(114,126)(115,131)(116,132)(117,133)(118,134)(119,127)(120,128)(121,129)(122,130)(139,143)(140,144)(141,145)(142,146)(147,159)(148,160)(149,161)(150,162)(151,167)(152,168)(153,169)(154,170)(155,163)(156,164)(157,165)(158,166)(175,179)(176,180)(177,181)(178,182)(183,195)(184,196)(185,197)(186,198)(187,203)(188,204)(189,205)(190,206)(191,199)(192,200)(193,201)(194,202)(211,215)(212,216)(213,217)(214,218);;
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, s2*s3*s4*s2*s3*s4*s2*s3*s4,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 3,113)( 4,114)( 5,111)( 6,112)( 7,117)( 8,118)( 9,115)( 10,116)( 11,121)( 12,122)( 13,119)( 14,120)( 15,125)( 16,126)( 17,123)( 18,124)( 19,129)( 20,130)( 21,127)( 22,128)( 23,133)( 24,134)( 25,131)( 26,132)( 27,137)( 28,138)( 29,135)( 30,136)( 31,141)( 32,142)( 33,139)( 34,140)( 35,145)( 36,146)( 37,143)( 38,144)( 39,149)( 40,150)( 41,147)( 42,148)( 43,153)( 44,154)( 45,151)( 46,152)( 47,157)( 48,158)( 49,155)( 50,156)( 51,161)( 52,162)( 53,159)( 54,160)( 55,165)( 56,166)( 57,163)( 58,164)( 59,169)( 60,170)( 61,167)( 62,168)( 63,173)( 64,174)( 65,171)( 66,172)( 67,177)( 68,178)( 69,175)( 70,176)( 71,181)( 72,182)( 73,179)( 74,180)( 75,185)( 76,186)( 77,183)( 78,184)( 79,189)( 80,190)( 81,187)( 82,188)( 83,193)( 84,194)( 85,191)( 86,192)( 87,197)( 88,198)( 89,195)( 90,196)( 91,201)( 92,202)( 93,199)( 94,200)( 95,205)( 96,206)( 97,203)( 98,204)( 99,209)(100,210)(101,207)(102,208)(103,213)(104,214)(105,211)(106,212)(107,217)(108,218)(109,215)(110,216);
s2 := Sym(218)!( 4, 5)( 7, 11)( 8, 13)( 9, 12)( 10, 14)( 16, 17)( 19, 23)( 20, 25)( 21, 24)( 22, 26)( 28, 29)( 31, 35)( 32, 37)( 33, 36)( 34, 38)( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 83)( 44, 85)( 45, 84)( 46, 86)( 47, 79)( 48, 81)( 49, 80)( 50, 82)( 51, 87)( 52, 89)( 53, 88)( 54, 90)( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)( 63, 99)( 64,101)( 65,100)( 66,102)( 67,107)( 68,109)( 69,108)( 70,110)( 71,103)( 72,105)( 73,104)( 74,106)(112,113)(115,119)(116,121)(117,120)(118,122)(124,125)(127,131)(128,133)(129,132)(130,134)(136,137)(139,143)(140,145)(141,144)(142,146)(147,183)(148,185)(149,184)(150,186)(151,191)(152,193)(153,192)(154,194)(155,187)(156,189)(157,188)(158,190)(159,195)(160,197)(161,196)(162,198)(163,203)(164,205)(165,204)(166,206)(167,199)(168,201)(169,200)(170,202)(171,207)(172,209)(173,208)(174,210)(175,215)(176,217)(177,216)(178,218)(179,211)(180,213)(181,212)(182,214);
s3 := Sym(218)!( 3, 75)( 4, 78)( 5, 77)( 6, 76)( 7, 79)( 8, 82)( 9, 81)( 10, 80)( 11, 83)( 12, 86)( 13, 85)( 14, 84)( 15,103)( 16,106)( 17,105)( 18,104)( 19,107)( 20,110)( 21,109)( 22,108)( 23, 99)( 24,102)( 25,101)( 26,100)( 27, 95)( 28, 98)( 29, 97)( 30, 96)( 31, 87)( 32, 90)( 33, 89)( 34, 88)( 35, 91)( 36, 94)( 37, 93)( 38, 92)( 40, 42)( 44, 46)( 48, 50)( 51, 67)( 52, 70)( 53, 69)( 54, 68)( 55, 71)( 56, 74)( 57, 73)( 58, 72)( 59, 63)( 60, 66)( 61, 65)( 62, 64)(111,183)(112,186)(113,185)(114,184)(115,187)(116,190)(117,189)(118,188)(119,191)(120,194)(121,193)(122,192)(123,211)(124,214)(125,213)(126,212)(127,215)(128,218)(129,217)(130,216)(131,207)(132,210)(133,209)(134,208)(135,203)(136,206)(137,205)(138,204)(139,195)(140,198)(141,197)(142,196)(143,199)(144,202)(145,201)(146,200)(148,150)(152,154)(156,158)(159,175)(160,178)(161,177)(162,176)(163,179)(164,182)(165,181)(166,180)(167,171)(168,174)(169,173)(170,172);
s4 := Sym(218)!( 3, 15)( 4, 16)( 5, 17)( 6, 18)( 7, 23)( 8, 24)( 9, 25)( 10, 26)( 11, 19)( 12, 20)( 13, 21)( 14, 22)( 31, 35)( 32, 36)( 33, 37)( 34, 38)( 39, 51)( 40, 52)( 41, 53)( 42, 54)( 43, 59)( 44, 60)( 45, 61)( 46, 62)( 47, 55)( 48, 56)( 49, 57)( 50, 58)( 67, 71)( 68, 72)( 69, 73)( 70, 74)( 75, 87)( 76, 88)( 77, 89)( 78, 90)( 79, 95)( 80, 96)( 81, 97)( 82, 98)( 83, 91)( 84, 92)( 85, 93)( 86, 94)(103,107)(104,108)(105,109)(106,110)(111,123)(112,124)(113,125)(114,126)(115,131)(116,132)(117,133)(118,134)(119,127)(120,128)(121,129)(122,130)(139,143)(140,144)(141,145)(142,146)(147,159)(148,160)(149,161)(150,162)(151,167)(152,168)(153,169)(154,170)(155,163)(156,164)(157,165)(158,166)(175,179)(176,180)(177,181)(178,182)(183,195)(184,196)(185,197)(186,198)(187,203)(188,204)(189,205)(190,206)(191,199)(192,200)(193,201)(194,202)(211,215)(212,216)(213,217)(214,218);
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,
s2*s3*s4*s2*s3*s4*s2*s3*s4, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope