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}*1728e
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 5
Schlafli Type : {2,4,6,6}
Number of vertices, edges, etc : 2, 4, 36, 54, 18
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
3-fold quotients : {2,4,6,6}*576d
9-fold quotients : {2,4,6,2}*192c
18-fold quotients : {2,4,3,2}*96
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3, 5)( 4, 6)( 7, 9)( 8, 10)( 11, 13)( 12, 14)( 15, 17)( 16, 18)
( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)( 32, 34)
( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)( 48, 50)
( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)( 64, 66)
( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)( 80, 82)
( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)( 96, 98)
( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)(112,114)
(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)(128,130)
(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)(144,146)
(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)(160,162)
(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)(176,178)
(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)(192,194)
(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)(208,210)
(211,213)(212,214)(215,217)(216,218);;
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 := ( 3,123)( 4,124)( 5,125)( 6,126)( 7,131)( 8,132)( 9,133)( 10,134)
( 11,127)( 12,128)( 13,129)( 14,130)( 15,111)( 16,112)( 17,113)( 18,114)
( 19,119)( 20,120)( 21,121)( 22,122)( 23,115)( 24,116)( 25,117)( 26,118)
( 27,135)( 28,136)( 29,137)( 30,138)( 31,143)( 32,144)( 33,145)( 34,146)
( 35,139)( 36,140)( 37,141)( 38,142)( 39,159)( 40,160)( 41,161)( 42,162)
( 43,167)( 44,168)( 45,169)( 46,170)( 47,163)( 48,164)( 49,165)( 50,166)
( 51,147)( 52,148)( 53,149)( 54,150)( 55,155)( 56,156)( 57,157)( 58,158)
( 59,151)( 60,152)( 61,153)( 62,154)( 63,171)( 64,172)( 65,173)( 66,174)
( 67,179)( 68,180)( 69,181)( 70,182)( 71,175)( 72,176)( 73,177)( 74,178)
( 75,195)( 76,196)( 77,197)( 78,198)( 79,203)( 80,204)( 81,205)( 82,206)
( 83,199)( 84,200)( 85,201)( 86,202)( 87,183)( 88,184)( 89,185)( 90,186)
( 91,191)( 92,192)( 93,193)( 94,194)( 95,187)( 96,188)( 97,189)( 98,190)
( 99,207)(100,208)(101,209)(102,210)(103,215)(104,216)(105,217)(106,218)
(107,211)(108,212)(109,213)(110,214);;
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*s1*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,
s4*s2*s3*s4*s2*s3*s2*s3*s4*s2*s3*s4*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 3, 5)( 4, 6)( 7, 9)( 8, 10)( 11, 13)( 12, 14)( 15, 17)
( 16, 18)( 19, 21)( 20, 22)( 23, 25)( 24, 26)( 27, 29)( 28, 30)( 31, 33)
( 32, 34)( 35, 37)( 36, 38)( 39, 41)( 40, 42)( 43, 45)( 44, 46)( 47, 49)
( 48, 50)( 51, 53)( 52, 54)( 55, 57)( 56, 58)( 59, 61)( 60, 62)( 63, 65)
( 64, 66)( 67, 69)( 68, 70)( 71, 73)( 72, 74)( 75, 77)( 76, 78)( 79, 81)
( 80, 82)( 83, 85)( 84, 86)( 87, 89)( 88, 90)( 91, 93)( 92, 94)( 95, 97)
( 96, 98)( 99,101)(100,102)(103,105)(104,106)(107,109)(108,110)(111,113)
(112,114)(115,117)(116,118)(119,121)(120,122)(123,125)(124,126)(127,129)
(128,130)(131,133)(132,134)(135,137)(136,138)(139,141)(140,142)(143,145)
(144,146)(147,149)(148,150)(151,153)(152,154)(155,157)(156,158)(159,161)
(160,162)(163,165)(164,166)(167,169)(168,170)(171,173)(172,174)(175,177)
(176,178)(179,181)(180,182)(183,185)(184,186)(187,189)(188,190)(191,193)
(192,194)(195,197)(196,198)(199,201)(200,202)(203,205)(204,206)(207,209)
(208,210)(211,213)(212,214)(215,217)(216,218);
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)!( 3,123)( 4,124)( 5,125)( 6,126)( 7,131)( 8,132)( 9,133)
( 10,134)( 11,127)( 12,128)( 13,129)( 14,130)( 15,111)( 16,112)( 17,113)
( 18,114)( 19,119)( 20,120)( 21,121)( 22,122)( 23,115)( 24,116)( 25,117)
( 26,118)( 27,135)( 28,136)( 29,137)( 30,138)( 31,143)( 32,144)( 33,145)
( 34,146)( 35,139)( 36,140)( 37,141)( 38,142)( 39,159)( 40,160)( 41,161)
( 42,162)( 43,167)( 44,168)( 45,169)( 46,170)( 47,163)( 48,164)( 49,165)
( 50,166)( 51,147)( 52,148)( 53,149)( 54,150)( 55,155)( 56,156)( 57,157)
( 58,158)( 59,151)( 60,152)( 61,153)( 62,154)( 63,171)( 64,172)( 65,173)
( 66,174)( 67,179)( 68,180)( 69,181)( 70,182)( 71,175)( 72,176)( 73,177)
( 74,178)( 75,195)( 76,196)( 77,197)( 78,198)( 79,203)( 80,204)( 81,205)
( 82,206)( 83,199)( 84,200)( 85,201)( 86,202)( 87,183)( 88,184)( 89,185)
( 90,186)( 91,191)( 92,192)( 93,193)( 94,194)( 95,187)( 96,188)( 97,189)
( 98,190)( 99,207)(100,208)(101,209)(102,210)(103,215)(104,216)(105,217)
(106,218)(107,211)(108,212)(109,213)(110,214);
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*s1*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,
s4*s2*s3*s4*s2*s3*s2*s3*s4*s2*s3*s4*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s2*s3 >;
to this polytope