Polytope of Type {2,6,3,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,3,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,46116)
Rank : 5
Schlafli Type : {2,6,3,4}
Number of vertices, edges, etc : 2, 18, 54, 36, 8
Order of s0s1s2s3s4 : 6
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 : {2,6,3,4}*864
3-fold quotients : {2,6,3,4}*576
4-fold quotients : {2,6,3,2}*432
6-fold quotients : {2,6,3,4}*288
9-fold quotients : {2,2,3,4}*192
12-fold quotients : {2,6,3,2}*144
18-fold quotients : {2,2,3,4}*96
36-fold quotients : {2,2,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 7, 11)( 8, 12)( 9, 13)( 10, 14)( 15, 27)( 16, 28)( 17, 29)( 18, 30)
( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 31)( 24, 32)( 25, 33)( 26, 34)
( 43, 47)( 44, 48)( 45, 49)( 46, 50)( 51, 63)( 52, 64)( 53, 65)( 54, 66)
( 55, 71)( 56, 72)( 57, 73)( 58, 74)( 59, 67)( 60, 68)( 61, 69)( 62, 70)
( 79, 83)( 80, 84)( 81, 85)( 82, 86)( 87, 99)( 88,100)( 89,101)( 90,102)
( 91,107)( 92,108)( 93,109)( 94,110)( 95,103)( 96,104)( 97,105)( 98,106)
(115,119)(116,120)(117,121)(118,122)(123,135)(124,136)(125,137)(126,138)
(127,143)(128,144)(129,145)(130,146)(131,139)(132,140)(133,141)(134,142)
(151,155)(152,156)(153,157)(154,158)(159,171)(160,172)(161,173)(162,174)
(163,179)(164,180)(165,181)(166,182)(167,175)(168,176)(169,177)(170,178)
(187,191)(188,192)(189,193)(190,194)(195,207)(196,208)(197,209)(198,210)
(199,215)(200,216)(201,217)(202,218)(203,211)(204,212)(205,213)(206,214);;
s2 := ( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 19)( 8, 21)( 9, 20)( 10, 22)
( 11, 23)( 12, 25)( 13, 24)( 14, 26)( 28, 29)( 32, 33)( 36, 37)( 39, 87)
( 40, 89)( 41, 88)( 42, 90)( 43, 91)( 44, 93)( 45, 92)( 46, 94)( 47, 95)
( 48, 97)( 49, 96)( 50, 98)( 51, 75)( 52, 77)( 53, 76)( 54, 78)( 55, 79)
( 56, 81)( 57, 80)( 58, 82)( 59, 83)( 60, 85)( 61, 84)( 62, 86)( 63, 99)
( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)( 69,104)( 70,106)( 71,107)
( 72,109)( 73,108)( 74,110)(111,123)(112,125)(113,124)(114,126)(115,127)
(116,129)(117,128)(118,130)(119,131)(120,133)(121,132)(122,134)(136,137)
(140,141)(144,145)(147,195)(148,197)(149,196)(150,198)(151,199)(152,201)
(153,200)(154,202)(155,203)(156,205)(157,204)(158,206)(159,183)(160,185)
(161,184)(162,186)(163,187)(164,189)(165,188)(166,190)(167,191)(168,193)
(169,192)(170,194)(171,207)(172,209)(173,208)(174,210)(175,211)(176,213)
(177,212)(178,214)(179,215)(180,217)(181,216)(182,218);;
s3 := ( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 43)( 8, 44)( 9, 46)( 10, 45)
( 11, 47)( 12, 48)( 13, 50)( 14, 49)( 15, 71)( 16, 72)( 17, 74)( 18, 73)
( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 67)( 24, 68)( 25, 70)( 26, 69)
( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 59)( 32, 60)( 33, 62)( 34, 61)
( 35, 51)( 36, 52)( 37, 54)( 38, 53)( 77, 78)( 81, 82)( 85, 86)( 87,107)
( 88,108)( 89,110)( 90,109)( 91, 99)( 92,100)( 93,102)( 94,101)( 95,103)
( 96,104)( 97,106)( 98,105)(111,147)(112,148)(113,150)(114,149)(115,151)
(116,152)(117,154)(118,153)(119,155)(120,156)(121,158)(122,157)(123,179)
(124,180)(125,182)(126,181)(127,171)(128,172)(129,174)(130,173)(131,175)
(132,176)(133,178)(134,177)(135,163)(136,164)(137,166)(138,165)(139,167)
(140,168)(141,170)(142,169)(143,159)(144,160)(145,162)(146,161)(185,186)
(189,190)(193,194)(195,215)(196,216)(197,218)(198,217)(199,207)(200,208)
(201,210)(202,209)(203,211)(204,212)(205,214)(206,213);;
s4 := ( 3,114)( 4,113)( 5,112)( 6,111)( 7,118)( 8,117)( 9,116)( 10,115)
( 11,122)( 12,121)( 13,120)( 14,119)( 15,126)( 16,125)( 17,124)( 18,123)
( 19,130)( 20,129)( 21,128)( 22,127)( 23,134)( 24,133)( 25,132)( 26,131)
( 27,138)( 28,137)( 29,136)( 30,135)( 31,142)( 32,141)( 33,140)( 34,139)
( 35,146)( 36,145)( 37,144)( 38,143)( 39,150)( 40,149)( 41,148)( 42,147)
( 43,154)( 44,153)( 45,152)( 46,151)( 47,158)( 48,157)( 49,156)( 50,155)
( 51,162)( 52,161)( 53,160)( 54,159)( 55,166)( 56,165)( 57,164)( 58,163)
( 59,170)( 60,169)( 61,168)( 62,167)( 63,174)( 64,173)( 65,172)( 66,171)
( 67,178)( 68,177)( 69,176)( 70,175)( 71,182)( 72,181)( 73,180)( 74,179)
( 75,186)( 76,185)( 77,184)( 78,183)( 79,190)( 80,189)( 81,188)( 82,187)
( 83,194)( 84,193)( 85,192)( 86,191)( 87,198)( 88,197)( 89,196)( 90,195)
( 91,202)( 92,201)( 93,200)( 94,199)( 95,206)( 96,205)( 97,204)( 98,203)
( 99,210)(100,209)(101,208)(102,207)(103,214)(104,213)(105,212)(106,211)
(107,218)(108,217)(109,216)(110,215);;
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,
s2*s3*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s4*s3*s2*s1*s2*s3*s4*s3*s2,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 7, 11)( 8, 12)( 9, 13)( 10, 14)( 15, 27)( 16, 28)( 17, 29)
( 18, 30)( 19, 35)( 20, 36)( 21, 37)( 22, 38)( 23, 31)( 24, 32)( 25, 33)
( 26, 34)( 43, 47)( 44, 48)( 45, 49)( 46, 50)( 51, 63)( 52, 64)( 53, 65)
( 54, 66)( 55, 71)( 56, 72)( 57, 73)( 58, 74)( 59, 67)( 60, 68)( 61, 69)
( 62, 70)( 79, 83)( 80, 84)( 81, 85)( 82, 86)( 87, 99)( 88,100)( 89,101)
( 90,102)( 91,107)( 92,108)( 93,109)( 94,110)( 95,103)( 96,104)( 97,105)
( 98,106)(115,119)(116,120)(117,121)(118,122)(123,135)(124,136)(125,137)
(126,138)(127,143)(128,144)(129,145)(130,146)(131,139)(132,140)(133,141)
(134,142)(151,155)(152,156)(153,157)(154,158)(159,171)(160,172)(161,173)
(162,174)(163,179)(164,180)(165,181)(166,182)(167,175)(168,176)(169,177)
(170,178)(187,191)(188,192)(189,193)(190,194)(195,207)(196,208)(197,209)
(198,210)(199,215)(200,216)(201,217)(202,218)(203,211)(204,212)(205,213)
(206,214);
s2 := Sym(218)!( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 19)( 8, 21)( 9, 20)
( 10, 22)( 11, 23)( 12, 25)( 13, 24)( 14, 26)( 28, 29)( 32, 33)( 36, 37)
( 39, 87)( 40, 89)( 41, 88)( 42, 90)( 43, 91)( 44, 93)( 45, 92)( 46, 94)
( 47, 95)( 48, 97)( 49, 96)( 50, 98)( 51, 75)( 52, 77)( 53, 76)( 54, 78)
( 55, 79)( 56, 81)( 57, 80)( 58, 82)( 59, 83)( 60, 85)( 61, 84)( 62, 86)
( 63, 99)( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)( 69,104)( 70,106)
( 71,107)( 72,109)( 73,108)( 74,110)(111,123)(112,125)(113,124)(114,126)
(115,127)(116,129)(117,128)(118,130)(119,131)(120,133)(121,132)(122,134)
(136,137)(140,141)(144,145)(147,195)(148,197)(149,196)(150,198)(151,199)
(152,201)(153,200)(154,202)(155,203)(156,205)(157,204)(158,206)(159,183)
(160,185)(161,184)(162,186)(163,187)(164,189)(165,188)(166,190)(167,191)
(168,193)(169,192)(170,194)(171,207)(172,209)(173,208)(174,210)(175,211)
(176,213)(177,212)(178,214)(179,215)(180,217)(181,216)(182,218);
s3 := Sym(218)!( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 43)( 8, 44)( 9, 46)
( 10, 45)( 11, 47)( 12, 48)( 13, 50)( 14, 49)( 15, 71)( 16, 72)( 17, 74)
( 18, 73)( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 67)( 24, 68)( 25, 70)
( 26, 69)( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 59)( 32, 60)( 33, 62)
( 34, 61)( 35, 51)( 36, 52)( 37, 54)( 38, 53)( 77, 78)( 81, 82)( 85, 86)
( 87,107)( 88,108)( 89,110)( 90,109)( 91, 99)( 92,100)( 93,102)( 94,101)
( 95,103)( 96,104)( 97,106)( 98,105)(111,147)(112,148)(113,150)(114,149)
(115,151)(116,152)(117,154)(118,153)(119,155)(120,156)(121,158)(122,157)
(123,179)(124,180)(125,182)(126,181)(127,171)(128,172)(129,174)(130,173)
(131,175)(132,176)(133,178)(134,177)(135,163)(136,164)(137,166)(138,165)
(139,167)(140,168)(141,170)(142,169)(143,159)(144,160)(145,162)(146,161)
(185,186)(189,190)(193,194)(195,215)(196,216)(197,218)(198,217)(199,207)
(200,208)(201,210)(202,209)(203,211)(204,212)(205,214)(206,213);
s4 := Sym(218)!( 3,114)( 4,113)( 5,112)( 6,111)( 7,118)( 8,117)( 9,116)
( 10,115)( 11,122)( 12,121)( 13,120)( 14,119)( 15,126)( 16,125)( 17,124)
( 18,123)( 19,130)( 20,129)( 21,128)( 22,127)( 23,134)( 24,133)( 25,132)
( 26,131)( 27,138)( 28,137)( 29,136)( 30,135)( 31,142)( 32,141)( 33,140)
( 34,139)( 35,146)( 36,145)( 37,144)( 38,143)( 39,150)( 40,149)( 41,148)
( 42,147)( 43,154)( 44,153)( 45,152)( 46,151)( 47,158)( 48,157)( 49,156)
( 50,155)( 51,162)( 52,161)( 53,160)( 54,159)( 55,166)( 56,165)( 57,164)
( 58,163)( 59,170)( 60,169)( 61,168)( 62,167)( 63,174)( 64,173)( 65,172)
( 66,171)( 67,178)( 68,177)( 69,176)( 70,175)( 71,182)( 72,181)( 73,180)
( 74,179)( 75,186)( 76,185)( 77,184)( 78,183)( 79,190)( 80,189)( 81,188)
( 82,187)( 83,194)( 84,193)( 85,192)( 86,191)( 87,198)( 88,197)( 89,196)
( 90,195)( 91,202)( 92,201)( 93,200)( 94,199)( 95,206)( 96,205)( 97,204)
( 98,203)( 99,210)(100,209)(101,208)(102,207)(103,214)(104,213)(105,212)
(106,211)(107,218)(108,217)(109,216)(110,215);
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, s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s4*s3*s2*s1*s2*s3*s4*s3*s2,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2 >;
to this polytope