Polytope of Type {6,4,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,36}*1728
Also Known As : {{6,4|2},{4,36|2}}. if this polytope has another name.
Group : SmallGroup(1728,14461)
Rank : 4
Schlafli Type : {6,4,36}
Number of vertices, edges, etc : 6, 12, 72, 36
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {6,2,36}*864, {6,4,18}*864
3-fold quotients : {2,4,36}*576a, {6,4,12}*576
4-fold quotients : {3,2,36}*432, {6,2,18}*432
6-fold quotients : {2,2,36}*288, {2,4,18}*288a, {6,2,12}*288, {6,4,6}*288
8-fold quotients : {3,2,18}*216, {6,2,9}*216
9-fold quotients : {2,4,12}*192a, {6,4,4}*192
12-fold quotients : {2,2,18}*144, {3,2,12}*144, {6,2,6}*144
16-fold quotients : {3,2,9}*108
18-fold quotients : {2,2,12}*96, {2,4,6}*96a, {6,2,4}*96, {6,4,2}*96a
24-fold quotients : {2,2,9}*72, {3,2,6}*72, {6,2,3}*72
27-fold quotients : {2,4,4}*64
36-fold quotients : {3,2,4}*48, {2,2,6}*48, {6,2,2}*48
48-fold quotients : {3,2,3}*36
54-fold quotients : {2,2,4}*32, {2,4,2}*32
72-fold quotients : {2,2,3}*24, {3,2,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)(173,182)
(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)(199,208)
(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216);;
s1 := ( 1, 10)( 2, 11)( 3, 12)( 4, 13)( 5, 14)( 6, 15)( 7, 16)( 8, 17)
( 9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)( 34, 43)
( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)( 86, 95)
( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,172)(110,173)(111,174)(112,175)
(113,176)(114,177)(115,178)(116,179)(117,180)(118,163)(119,164)(120,165)
(121,166)(122,167)(123,168)(124,169)(125,170)(126,171)(127,181)(128,182)
(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,199)
(137,200)(138,201)(139,202)(140,203)(141,204)(142,205)(143,206)(144,207)
(145,190)(146,191)(147,192)(148,193)(149,194)(150,195)(151,196)(152,197)
(153,198)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)
(161,215)(162,216);;
s2 := ( 1,109)( 2,111)( 3,110)( 4,117)( 5,116)( 6,115)( 7,114)( 8,113)
( 9,112)( 10,118)( 11,120)( 12,119)( 13,126)( 14,125)( 15,124)( 16,123)
( 17,122)( 18,121)( 19,127)( 20,129)( 21,128)( 22,135)( 23,134)( 24,133)
( 25,132)( 26,131)( 27,130)( 28,136)( 29,138)( 30,137)( 31,144)( 32,143)
( 33,142)( 34,141)( 35,140)( 36,139)( 37,145)( 38,147)( 39,146)( 40,153)
( 41,152)( 42,151)( 43,150)( 44,149)( 45,148)( 46,154)( 47,156)( 48,155)
( 49,162)( 50,161)( 51,160)( 52,159)( 53,158)( 54,157)( 55,163)( 56,165)
( 57,164)( 58,171)( 59,170)( 60,169)( 61,168)( 62,167)( 63,166)( 64,172)
( 65,174)( 66,173)( 67,180)( 68,179)( 69,178)( 70,177)( 71,176)( 72,175)
( 73,181)( 74,183)( 75,182)( 76,189)( 77,188)( 78,187)( 79,186)( 80,185)
( 81,184)( 82,190)( 83,192)( 84,191)( 85,198)( 86,197)( 87,196)( 88,195)
( 89,194)( 90,193)( 91,199)( 92,201)( 93,200)( 94,207)( 95,206)( 96,205)
( 97,204)( 98,203)( 99,202)(100,208)(101,210)(102,209)(103,216)(104,215)
(105,214)(106,213)(107,212)(108,211);;
s3 := ( 1, 4)( 2, 6)( 3, 5)( 7, 9)( 10, 13)( 11, 15)( 12, 14)( 16, 18)
( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 31)( 29, 33)( 30, 32)( 34, 36)
( 37, 40)( 38, 42)( 39, 41)( 43, 45)( 46, 49)( 47, 51)( 48, 50)( 52, 54)
( 55, 58)( 56, 60)( 57, 59)( 61, 63)( 64, 67)( 65, 69)( 66, 68)( 70, 72)
( 73, 76)( 74, 78)( 75, 77)( 79, 81)( 82, 85)( 83, 87)( 84, 86)( 88, 90)
( 91, 94)( 92, 96)( 93, 95)( 97, 99)(100,103)(101,105)(102,104)(106,108)
(109,139)(110,141)(111,140)(112,136)(113,138)(114,137)(115,144)(116,143)
(117,142)(118,148)(119,150)(120,149)(121,145)(122,147)(123,146)(124,153)
(125,152)(126,151)(127,157)(128,159)(129,158)(130,154)(131,156)(132,155)
(133,162)(134,161)(135,160)(163,193)(164,195)(165,194)(166,190)(167,192)
(168,191)(169,198)(170,197)(171,196)(172,202)(173,204)(174,203)(175,199)
(176,201)(177,200)(178,207)(179,206)(180,205)(181,211)(182,213)(183,212)
(184,208)(185,210)(186,209)(187,216)(188,215)(189,214);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)
(173,182)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)
(199,208)(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)
(207,216);
s1 := Sym(216)!( 1, 10)( 2, 11)( 3, 12)( 4, 13)( 5, 14)( 6, 15)( 7, 16)
( 8, 17)( 9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)
( 34, 43)( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)
( 60, 69)( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)
( 86, 95)( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,172)(110,173)(111,174)
(112,175)(113,176)(114,177)(115,178)(116,179)(117,180)(118,163)(119,164)
(120,165)(121,166)(122,167)(123,168)(124,169)(125,170)(126,171)(127,181)
(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)
(136,199)(137,200)(138,201)(139,202)(140,203)(141,204)(142,205)(143,206)
(144,207)(145,190)(146,191)(147,192)(148,193)(149,194)(150,195)(151,196)
(152,197)(153,198)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)
(160,214)(161,215)(162,216);
s2 := Sym(216)!( 1,109)( 2,111)( 3,110)( 4,117)( 5,116)( 6,115)( 7,114)
( 8,113)( 9,112)( 10,118)( 11,120)( 12,119)( 13,126)( 14,125)( 15,124)
( 16,123)( 17,122)( 18,121)( 19,127)( 20,129)( 21,128)( 22,135)( 23,134)
( 24,133)( 25,132)( 26,131)( 27,130)( 28,136)( 29,138)( 30,137)( 31,144)
( 32,143)( 33,142)( 34,141)( 35,140)( 36,139)( 37,145)( 38,147)( 39,146)
( 40,153)( 41,152)( 42,151)( 43,150)( 44,149)( 45,148)( 46,154)( 47,156)
( 48,155)( 49,162)( 50,161)( 51,160)( 52,159)( 53,158)( 54,157)( 55,163)
( 56,165)( 57,164)( 58,171)( 59,170)( 60,169)( 61,168)( 62,167)( 63,166)
( 64,172)( 65,174)( 66,173)( 67,180)( 68,179)( 69,178)( 70,177)( 71,176)
( 72,175)( 73,181)( 74,183)( 75,182)( 76,189)( 77,188)( 78,187)( 79,186)
( 80,185)( 81,184)( 82,190)( 83,192)( 84,191)( 85,198)( 86,197)( 87,196)
( 88,195)( 89,194)( 90,193)( 91,199)( 92,201)( 93,200)( 94,207)( 95,206)
( 96,205)( 97,204)( 98,203)( 99,202)(100,208)(101,210)(102,209)(103,216)
(104,215)(105,214)(106,213)(107,212)(108,211);
s3 := Sym(216)!( 1, 4)( 2, 6)( 3, 5)( 7, 9)( 10, 13)( 11, 15)( 12, 14)
( 16, 18)( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 31)( 29, 33)( 30, 32)
( 34, 36)( 37, 40)( 38, 42)( 39, 41)( 43, 45)( 46, 49)( 47, 51)( 48, 50)
( 52, 54)( 55, 58)( 56, 60)( 57, 59)( 61, 63)( 64, 67)( 65, 69)( 66, 68)
( 70, 72)( 73, 76)( 74, 78)( 75, 77)( 79, 81)( 82, 85)( 83, 87)( 84, 86)
( 88, 90)( 91, 94)( 92, 96)( 93, 95)( 97, 99)(100,103)(101,105)(102,104)
(106,108)(109,139)(110,141)(111,140)(112,136)(113,138)(114,137)(115,144)
(116,143)(117,142)(118,148)(119,150)(120,149)(121,145)(122,147)(123,146)
(124,153)(125,152)(126,151)(127,157)(128,159)(129,158)(130,154)(131,156)
(132,155)(133,162)(134,161)(135,160)(163,193)(164,195)(165,194)(166,190)
(167,192)(168,191)(169,198)(170,197)(171,196)(172,202)(173,204)(174,203)
(175,199)(176,201)(177,200)(178,207)(179,206)(180,205)(181,211)(182,213)
(183,212)(184,208)(185,210)(186,209)(187,216)(188,215)(189,214);
poly := sub<Sym(216)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope