Polytope of Type {2,36,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,36,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,46114)
Rank : 4
Schlafli Type : {2,36,6}
Number of vertices, edges, etc : 2, 72, 216, 12
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 : {2,36,6}*864c
3-fold quotients : {2,12,6}*576a
4-fold quotients : {2,18,6}*432a
6-fold quotients : {2,12,6}*288d
9-fold quotients : {2,4,6}*192
12-fold quotients : {2,18,2}*144, {2,6,6}*144a
18-fold quotients : {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
24-fold quotients : {2,9,2}*72
36-fold quotients : {2,4,3}*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 := ( 3, 5)( 4, 6)( 7, 13)( 8, 14)( 9, 11)( 10, 12)( 15, 33)( 16, 34)
( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 27)( 22, 28)( 23, 37)( 24, 38)
( 25, 35)( 26, 36)( 39, 41)( 40, 42)( 43, 49)( 44, 50)( 45, 47)( 46, 48)
( 51, 69)( 52, 70)( 53, 67)( 54, 68)( 55, 65)( 56, 66)( 57, 63)( 58, 64)
( 59, 73)( 60, 74)( 61, 71)( 62, 72)( 75, 77)( 76, 78)( 79, 85)( 80, 86)
( 81, 83)( 82, 84)( 87,105)( 88,106)( 89,103)( 90,104)( 91,101)( 92,102)
( 93, 99)( 94,100)( 95,109)( 96,110)( 97,107)( 98,108)(111,113)(112,114)
(115,121)(116,122)(117,119)(118,120)(123,141)(124,142)(125,139)(126,140)
(127,137)(128,138)(129,135)(130,136)(131,145)(132,146)(133,143)(134,144)
(147,149)(148,150)(151,157)(152,158)(153,155)(154,156)(159,177)(160,178)
(161,175)(162,176)(163,173)(164,174)(165,171)(166,172)(167,181)(168,182)
(169,179)(170,180)(183,185)(184,186)(187,193)(188,194)(189,191)(190,192)
(195,213)(196,214)(197,211)(198,212)(199,209)(200,210)(201,207)(202,208)
(203,217)(204,218)(205,215)(206,216);;
s2 := ( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 23)( 8, 25)( 9, 24)( 10, 26)
( 11, 19)( 12, 21)( 13, 20)( 14, 22)( 27, 31)( 28, 33)( 29, 32)( 30, 34)
( 36, 37)( 39, 87)( 40, 89)( 41, 88)( 42, 90)( 43, 95)( 44, 97)( 45, 96)
( 46, 98)( 47, 91)( 48, 93)( 49, 92)( 50, 94)( 51, 75)( 52, 77)( 53, 76)
( 54, 78)( 55, 83)( 56, 85)( 57, 84)( 58, 86)( 59, 79)( 60, 81)( 61, 80)
( 62, 82)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)( 69,100)
( 70,102)( 71,107)( 72,109)( 73,108)( 74,110)(111,123)(112,125)(113,124)
(114,126)(115,131)(116,133)(117,132)(118,134)(119,127)(120,129)(121,128)
(122,130)(135,139)(136,141)(137,140)(138,142)(144,145)(147,195)(148,197)
(149,196)(150,198)(151,203)(152,205)(153,204)(154,206)(155,199)(156,201)
(157,200)(158,202)(159,183)(160,185)(161,184)(162,186)(163,191)(164,193)
(165,192)(166,194)(167,187)(168,189)(169,188)(170,190)(171,211)(172,213)
(173,212)(174,214)(175,207)(176,209)(177,208)(178,210)(179,215)(180,217)
(181,216)(182,218);;
s3 := ( 3,183)( 4,186)( 5,185)( 6,184)( 7,187)( 8,190)( 9,189)( 10,188)
( 11,191)( 12,194)( 13,193)( 14,192)( 15,195)( 16,198)( 17,197)( 18,196)
( 19,199)( 20,202)( 21,201)( 22,200)( 23,203)( 24,206)( 25,205)( 26,204)
( 27,207)( 28,210)( 29,209)( 30,208)( 31,211)( 32,214)( 33,213)( 34,212)
( 35,215)( 36,218)( 37,217)( 38,216)( 39,147)( 40,150)( 41,149)( 42,148)
( 43,151)( 44,154)( 45,153)( 46,152)( 47,155)( 48,158)( 49,157)( 50,156)
( 51,159)( 52,162)( 53,161)( 54,160)( 55,163)( 56,166)( 57,165)( 58,164)
( 59,167)( 60,170)( 61,169)( 62,168)( 63,171)( 64,174)( 65,173)( 66,172)
( 67,175)( 68,178)( 69,177)( 70,176)( 71,179)( 72,182)( 73,181)( 74,180)
( 75,111)( 76,114)( 77,113)( 78,112)( 79,115)( 80,118)( 81,117)( 82,116)
( 83,119)( 84,122)( 85,121)( 86,120)( 87,123)( 88,126)( 89,125)( 90,124)
( 91,127)( 92,130)( 93,129)( 94,128)( 95,131)( 96,134)( 97,133)( 98,132)
( 99,135)(100,138)(101,137)(102,136)(103,139)(104,142)(105,141)(106,140)
(107,143)(108,146)(109,145)(110,144);;
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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(218)!(1,2);
s1 := Sym(218)!( 3, 5)( 4, 6)( 7, 13)( 8, 14)( 9, 11)( 10, 12)( 15, 33)
( 16, 34)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 27)( 22, 28)( 23, 37)
( 24, 38)( 25, 35)( 26, 36)( 39, 41)( 40, 42)( 43, 49)( 44, 50)( 45, 47)
( 46, 48)( 51, 69)( 52, 70)( 53, 67)( 54, 68)( 55, 65)( 56, 66)( 57, 63)
( 58, 64)( 59, 73)( 60, 74)( 61, 71)( 62, 72)( 75, 77)( 76, 78)( 79, 85)
( 80, 86)( 81, 83)( 82, 84)( 87,105)( 88,106)( 89,103)( 90,104)( 91,101)
( 92,102)( 93, 99)( 94,100)( 95,109)( 96,110)( 97,107)( 98,108)(111,113)
(112,114)(115,121)(116,122)(117,119)(118,120)(123,141)(124,142)(125,139)
(126,140)(127,137)(128,138)(129,135)(130,136)(131,145)(132,146)(133,143)
(134,144)(147,149)(148,150)(151,157)(152,158)(153,155)(154,156)(159,177)
(160,178)(161,175)(162,176)(163,173)(164,174)(165,171)(166,172)(167,181)
(168,182)(169,179)(170,180)(183,185)(184,186)(187,193)(188,194)(189,191)
(190,192)(195,213)(196,214)(197,211)(198,212)(199,209)(200,210)(201,207)
(202,208)(203,217)(204,218)(205,215)(206,216);
s2 := Sym(218)!( 3, 15)( 4, 17)( 5, 16)( 6, 18)( 7, 23)( 8, 25)( 9, 24)
( 10, 26)( 11, 19)( 12, 21)( 13, 20)( 14, 22)( 27, 31)( 28, 33)( 29, 32)
( 30, 34)( 36, 37)( 39, 87)( 40, 89)( 41, 88)( 42, 90)( 43, 95)( 44, 97)
( 45, 96)( 46, 98)( 47, 91)( 48, 93)( 49, 92)( 50, 94)( 51, 75)( 52, 77)
( 53, 76)( 54, 78)( 55, 83)( 56, 85)( 57, 84)( 58, 86)( 59, 79)( 60, 81)
( 61, 80)( 62, 82)( 63,103)( 64,105)( 65,104)( 66,106)( 67, 99)( 68,101)
( 69,100)( 70,102)( 71,107)( 72,109)( 73,108)( 74,110)(111,123)(112,125)
(113,124)(114,126)(115,131)(116,133)(117,132)(118,134)(119,127)(120,129)
(121,128)(122,130)(135,139)(136,141)(137,140)(138,142)(144,145)(147,195)
(148,197)(149,196)(150,198)(151,203)(152,205)(153,204)(154,206)(155,199)
(156,201)(157,200)(158,202)(159,183)(160,185)(161,184)(162,186)(163,191)
(164,193)(165,192)(166,194)(167,187)(168,189)(169,188)(170,190)(171,211)
(172,213)(173,212)(174,214)(175,207)(176,209)(177,208)(178,210)(179,215)
(180,217)(181,216)(182,218);
s3 := Sym(218)!( 3,183)( 4,186)( 5,185)( 6,184)( 7,187)( 8,190)( 9,189)
( 10,188)( 11,191)( 12,194)( 13,193)( 14,192)( 15,195)( 16,198)( 17,197)
( 18,196)( 19,199)( 20,202)( 21,201)( 22,200)( 23,203)( 24,206)( 25,205)
( 26,204)( 27,207)( 28,210)( 29,209)( 30,208)( 31,211)( 32,214)( 33,213)
( 34,212)( 35,215)( 36,218)( 37,217)( 38,216)( 39,147)( 40,150)( 41,149)
( 42,148)( 43,151)( 44,154)( 45,153)( 46,152)( 47,155)( 48,158)( 49,157)
( 50,156)( 51,159)( 52,162)( 53,161)( 54,160)( 55,163)( 56,166)( 57,165)
( 58,164)( 59,167)( 60,170)( 61,169)( 62,168)( 63,171)( 64,174)( 65,173)
( 66,172)( 67,175)( 68,178)( 69,177)( 70,176)( 71,179)( 72,182)( 73,181)
( 74,180)( 75,111)( 76,114)( 77,113)( 78,112)( 79,115)( 80,118)( 81,117)
( 82,116)( 83,119)( 84,122)( 85,121)( 86,120)( 87,123)( 88,126)( 89,125)
( 90,124)( 91,127)( 92,130)( 93,129)( 94,128)( 95,131)( 96,134)( 97,133)
( 98,132)( 99,135)(100,138)(101,137)(102,136)(103,139)(104,142)(105,141)
(106,140)(107,143)(108,146)(109,145)(110,144);
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2 >;
to this polytope