Polytope of Type {2,6,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,36}*1728
if this polytope has a name.
Group : SmallGroup(1728,46114)
Rank : 4
Schlafli Type : {2,6,36}
Number of vertices, edges, etc : 2, 12, 216, 72
Order of s0s1s2s3 : 18
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) :
2-fold quotients : {2,6,36}*864c
3-fold quotients : {2,6,12}*576a
4-fold quotients : {2,6,18}*432a
6-fold quotients : {2,6,12}*288d
9-fold quotients : {2,6,4}*192
12-fold quotients : {2,2,18}*144, {2,6,6}*144a
18-fold quotients : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
24-fold quotients : {2,2,9}*72
36-fold quotients : {2,3,4}*48, {2,2,6}*48, {2,6,2}*48
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 8, 9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)( 32, 33)
( 36, 37)( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)
( 46, 82)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51, 87)( 52, 89)( 53, 88)
( 54, 90)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 95)( 60, 97)( 61, 96)
( 62, 98)( 63, 99)( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)( 69,104)
( 70,106)( 71,107)( 72,109)( 73,108)( 74,110)(112,113)(116,117)(120,121)
(124,125)(128,129)(132,133)(136,137)(140,141)(144,145)(147,183)(148,185)
(149,184)(150,186)(151,187)(152,189)(153,188)(154,190)(155,191)(156,193)
(157,192)(158,194)(159,195)(160,197)(161,196)(162,198)(163,199)(164,201)
(165,200)(166,202)(167,203)(168,205)(169,204)(170,206)(171,207)(172,209)
(173,208)(174,210)(175,211)(176,213)(177,212)(178,214)(179,215)(180,217)
(181,216)(182,218);;
s2 := ( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)( 10, 49)
( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 67)( 16, 68)( 17, 70)( 18, 69)
( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 71)( 24, 72)( 25, 74)( 26, 73)
( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 51)( 32, 52)( 33, 54)( 34, 53)
( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 77, 78)( 79, 83)( 80, 84)( 81, 86)
( 82, 85)( 87,103)( 88,104)( 89,106)( 90,105)( 91, 99)( 92,100)( 93,102)
( 94,101)( 95,107)( 96,108)( 97,110)( 98,109)(111,147)(112,148)(113,150)
(114,149)(115,155)(116,156)(117,158)(118,157)(119,151)(120,152)(121,154)
(122,153)(123,175)(124,176)(125,178)(126,177)(127,171)(128,172)(129,174)
(130,173)(131,179)(132,180)(133,182)(134,181)(135,163)(136,164)(137,166)
(138,165)(139,159)(140,160)(141,162)(142,161)(143,167)(144,168)(145,170)
(146,169)(185,186)(187,191)(188,192)(189,194)(190,193)(195,211)(196,212)
(197,214)(198,213)(199,207)(200,208)(201,210)(202,209)(203,215)(204,216)
(205,218)(206,217);;
s3 := ( 3,126)( 4,125)( 5,124)( 6,123)( 7,134)( 8,133)( 9,132)( 10,131)
( 11,130)( 12,129)( 13,128)( 14,127)( 15,114)( 16,113)( 17,112)( 18,111)
( 19,122)( 20,121)( 21,120)( 22,119)( 23,118)( 24,117)( 25,116)( 26,115)
( 27,142)( 28,141)( 29,140)( 30,139)( 31,138)( 32,137)( 33,136)( 34,135)
( 35,146)( 36,145)( 37,144)( 38,143)( 39,162)( 40,161)( 41,160)( 42,159)
( 43,170)( 44,169)( 45,168)( 46,167)( 47,166)( 48,165)( 49,164)( 50,163)
( 51,150)( 52,149)( 53,148)( 54,147)( 55,158)( 56,157)( 57,156)( 58,155)
( 59,154)( 60,153)( 61,152)( 62,151)( 63,178)( 64,177)( 65,176)( 66,175)
( 67,174)( 68,173)( 69,172)( 70,171)( 71,182)( 72,181)( 73,180)( 74,179)
( 75,198)( 76,197)( 77,196)( 78,195)( 79,206)( 80,205)( 81,204)( 82,203)
( 83,202)( 84,201)( 85,200)( 86,199)( 87,186)( 88,185)( 89,184)( 90,183)
( 91,194)( 92,193)( 93,192)( 94,191)( 95,190)( 96,189)( 97,188)( 98,187)
( 99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)(106,207)
(107,218)(108,217)(109,216)(110,215);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 4, 5)( 8, 9)( 12, 13)( 16, 17)( 20, 21)( 24, 25)( 28, 29)
( 32, 33)( 36, 37)( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)
( 45, 80)( 46, 82)( 47, 83)( 48, 85)( 49, 84)( 50, 86)( 51, 87)( 52, 89)
( 53, 88)( 54, 90)( 55, 91)( 56, 93)( 57, 92)( 58, 94)( 59, 95)( 60, 97)
( 61, 96)( 62, 98)( 63, 99)( 64,101)( 65,100)( 66,102)( 67,103)( 68,105)
( 69,104)( 70,106)( 71,107)( 72,109)( 73,108)( 74,110)(112,113)(116,117)
(120,121)(124,125)(128,129)(132,133)(136,137)(140,141)(144,145)(147,183)
(148,185)(149,184)(150,186)(151,187)(152,189)(153,188)(154,190)(155,191)
(156,193)(157,192)(158,194)(159,195)(160,197)(161,196)(162,198)(163,199)
(164,201)(165,200)(166,202)(167,203)(168,205)(169,204)(170,206)(171,207)
(172,209)(173,208)(174,210)(175,211)(176,213)(177,212)(178,214)(179,215)
(180,217)(181,216)(182,218);
s2 := Sym(218)!( 3, 39)( 4, 40)( 5, 42)( 6, 41)( 7, 47)( 8, 48)( 9, 50)
( 10, 49)( 11, 43)( 12, 44)( 13, 46)( 14, 45)( 15, 67)( 16, 68)( 17, 70)
( 18, 69)( 19, 63)( 20, 64)( 21, 66)( 22, 65)( 23, 71)( 24, 72)( 25, 74)
( 26, 73)( 27, 55)( 28, 56)( 29, 58)( 30, 57)( 31, 51)( 32, 52)( 33, 54)
( 34, 53)( 35, 59)( 36, 60)( 37, 62)( 38, 61)( 77, 78)( 79, 83)( 80, 84)
( 81, 86)( 82, 85)( 87,103)( 88,104)( 89,106)( 90,105)( 91, 99)( 92,100)
( 93,102)( 94,101)( 95,107)( 96,108)( 97,110)( 98,109)(111,147)(112,148)
(113,150)(114,149)(115,155)(116,156)(117,158)(118,157)(119,151)(120,152)
(121,154)(122,153)(123,175)(124,176)(125,178)(126,177)(127,171)(128,172)
(129,174)(130,173)(131,179)(132,180)(133,182)(134,181)(135,163)(136,164)
(137,166)(138,165)(139,159)(140,160)(141,162)(142,161)(143,167)(144,168)
(145,170)(146,169)(185,186)(187,191)(188,192)(189,194)(190,193)(195,211)
(196,212)(197,214)(198,213)(199,207)(200,208)(201,210)(202,209)(203,215)
(204,216)(205,218)(206,217);
s3 := Sym(218)!( 3,126)( 4,125)( 5,124)( 6,123)( 7,134)( 8,133)( 9,132)
( 10,131)( 11,130)( 12,129)( 13,128)( 14,127)( 15,114)( 16,113)( 17,112)
( 18,111)( 19,122)( 20,121)( 21,120)( 22,119)( 23,118)( 24,117)( 25,116)
( 26,115)( 27,142)( 28,141)( 29,140)( 30,139)( 31,138)( 32,137)( 33,136)
( 34,135)( 35,146)( 36,145)( 37,144)( 38,143)( 39,162)( 40,161)( 41,160)
( 42,159)( 43,170)( 44,169)( 45,168)( 46,167)( 47,166)( 48,165)( 49,164)
( 50,163)( 51,150)( 52,149)( 53,148)( 54,147)( 55,158)( 56,157)( 57,156)
( 58,155)( 59,154)( 60,153)( 61,152)( 62,151)( 63,178)( 64,177)( 65,176)
( 66,175)( 67,174)( 68,173)( 69,172)( 70,171)( 71,182)( 72,181)( 73,180)
( 74,179)( 75,198)( 76,197)( 77,196)( 78,195)( 79,206)( 80,205)( 81,204)
( 82,203)( 83,202)( 84,201)( 85,200)( 86,199)( 87,186)( 88,185)( 89,184)
( 90,183)( 91,194)( 92,193)( 93,192)( 94,191)( 95,190)( 96,189)( 97,188)
( 98,187)( 99,214)(100,213)(101,212)(102,211)(103,210)(104,209)(105,208)
(106,207)(107,218)(108,217)(109,216)(110,215);
poly := sub<Sym(218)|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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2 >;
to this polytope