Polytope of Type {2,2,4,27}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,4,27}*1728
if this polytope has a name.
Group : SmallGroup(1728,20782)
Rank : 5
Schlafli Type : {2,2,4,27}
Number of vertices, edges, etc : 2, 2, 8, 108, 54
Order of s0s1s2s3s4 : 54
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 : {2,2,4,27}*864
3-fold quotients : {2,2,4,9}*576
4-fold quotients : {2,2,2,27}*432
6-fold quotients : {2,2,4,9}*288
9-fold quotients : {2,2,4,3}*192
12-fold quotients : {2,2,2,9}*144
18-fold quotients : {2,2,4,3}*96
36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 5,115)( 6,116)( 7,113)( 8,114)( 9,119)( 10,120)( 11,117)( 12,118)
( 13,123)( 14,124)( 15,121)( 16,122)( 17,127)( 18,128)( 19,125)( 20,126)
( 21,131)( 22,132)( 23,129)( 24,130)( 25,135)( 26,136)( 27,133)( 28,134)
( 29,139)( 30,140)( 31,137)( 32,138)( 33,143)( 34,144)( 35,141)( 36,142)
( 37,147)( 38,148)( 39,145)( 40,146)( 41,151)( 42,152)( 43,149)( 44,150)
( 45,155)( 46,156)( 47,153)( 48,154)( 49,159)( 50,160)( 51,157)( 52,158)
( 53,163)( 54,164)( 55,161)( 56,162)( 57,167)( 58,168)( 59,165)( 60,166)
( 61,171)( 62,172)( 63,169)( 64,170)( 65,175)( 66,176)( 67,173)( 68,174)
( 69,179)( 70,180)( 71,177)( 72,178)( 73,183)( 74,184)( 75,181)( 76,182)
( 77,187)( 78,188)( 79,185)( 80,186)( 81,191)( 82,192)( 83,189)( 84,190)
( 85,195)( 86,196)( 87,193)( 88,194)( 89,199)( 90,200)( 91,197)( 92,198)
( 93,203)( 94,204)( 95,201)( 96,202)( 97,207)( 98,208)( 99,205)(100,206)
(101,211)(102,212)(103,209)(104,210)(105,215)(106,216)(107,213)(108,214)
(109,219)(110,220)(111,217)(112,218);;
s3 := ( 6, 7)( 9, 13)( 10, 15)( 11, 14)( 12, 16)( 17, 33)( 18, 35)( 19, 34)
( 20, 36)( 21, 29)( 22, 31)( 23, 30)( 24, 32)( 25, 37)( 26, 39)( 27, 38)
( 28, 40)( 41, 89)( 42, 91)( 43, 90)( 44, 92)( 45, 97)( 46, 99)( 47, 98)
( 48,100)( 49, 93)( 50, 95)( 51, 94)( 52, 96)( 53, 77)( 54, 79)( 55, 78)
( 56, 80)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 81)( 62, 83)( 63, 82)
( 64, 84)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)( 71,102)
( 72,104)( 73,109)( 74,111)( 75,110)( 76,112)(114,115)(117,121)(118,123)
(119,122)(120,124)(125,141)(126,143)(127,142)(128,144)(129,137)(130,139)
(131,138)(132,140)(133,145)(134,147)(135,146)(136,148)(149,197)(150,199)
(151,198)(152,200)(153,205)(154,207)(155,206)(156,208)(157,201)(158,203)
(159,202)(160,204)(161,185)(162,187)(163,186)(164,188)(165,193)(166,195)
(167,194)(168,196)(169,189)(170,191)(171,190)(172,192)(173,213)(174,215)
(175,214)(176,216)(177,209)(178,211)(179,210)(180,212)(181,217)(182,219)
(183,218)(184,220);;
s4 := ( 5, 77)( 6, 80)( 7, 79)( 8, 78)( 9, 85)( 10, 88)( 11, 87)( 12, 86)
( 13, 81)( 14, 84)( 15, 83)( 16, 82)( 17,105)( 18,108)( 19,107)( 20,106)
( 21,101)( 22,104)( 23,103)( 24,102)( 25,109)( 26,112)( 27,111)( 28,110)
( 29, 93)( 30, 96)( 31, 95)( 32, 94)( 33, 89)( 34, 92)( 35, 91)( 36, 90)
( 37, 97)( 38,100)( 39, 99)( 40, 98)( 42, 44)( 45, 49)( 46, 52)( 47, 51)
( 48, 50)( 53, 69)( 54, 72)( 55, 71)( 56, 70)( 57, 65)( 58, 68)( 59, 67)
( 60, 66)( 61, 73)( 62, 76)( 63, 75)( 64, 74)(113,185)(114,188)(115,187)
(116,186)(117,193)(118,196)(119,195)(120,194)(121,189)(122,192)(123,191)
(124,190)(125,213)(126,216)(127,215)(128,214)(129,209)(130,212)(131,211)
(132,210)(133,217)(134,220)(135,219)(136,218)(137,201)(138,204)(139,203)
(140,202)(141,197)(142,200)(143,199)(144,198)(145,205)(146,208)(147,207)
(148,206)(150,152)(153,157)(154,160)(155,159)(156,158)(161,177)(162,180)
(163,179)(164,178)(165,173)(166,176)(167,175)(168,174)(169,181)(170,184)
(171,183)(172,182);;
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, s1*s2*s1*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*s2*s3,
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(220)!(1,2);
s1 := Sym(220)!(3,4);
s2 := Sym(220)!( 5,115)( 6,116)( 7,113)( 8,114)( 9,119)( 10,120)( 11,117)
( 12,118)( 13,123)( 14,124)( 15,121)( 16,122)( 17,127)( 18,128)( 19,125)
( 20,126)( 21,131)( 22,132)( 23,129)( 24,130)( 25,135)( 26,136)( 27,133)
( 28,134)( 29,139)( 30,140)( 31,137)( 32,138)( 33,143)( 34,144)( 35,141)
( 36,142)( 37,147)( 38,148)( 39,145)( 40,146)( 41,151)( 42,152)( 43,149)
( 44,150)( 45,155)( 46,156)( 47,153)( 48,154)( 49,159)( 50,160)( 51,157)
( 52,158)( 53,163)( 54,164)( 55,161)( 56,162)( 57,167)( 58,168)( 59,165)
( 60,166)( 61,171)( 62,172)( 63,169)( 64,170)( 65,175)( 66,176)( 67,173)
( 68,174)( 69,179)( 70,180)( 71,177)( 72,178)( 73,183)( 74,184)( 75,181)
( 76,182)( 77,187)( 78,188)( 79,185)( 80,186)( 81,191)( 82,192)( 83,189)
( 84,190)( 85,195)( 86,196)( 87,193)( 88,194)( 89,199)( 90,200)( 91,197)
( 92,198)( 93,203)( 94,204)( 95,201)( 96,202)( 97,207)( 98,208)( 99,205)
(100,206)(101,211)(102,212)(103,209)(104,210)(105,215)(106,216)(107,213)
(108,214)(109,219)(110,220)(111,217)(112,218);
s3 := Sym(220)!( 6, 7)( 9, 13)( 10, 15)( 11, 14)( 12, 16)( 17, 33)( 18, 35)
( 19, 34)( 20, 36)( 21, 29)( 22, 31)( 23, 30)( 24, 32)( 25, 37)( 26, 39)
( 27, 38)( 28, 40)( 41, 89)( 42, 91)( 43, 90)( 44, 92)( 45, 97)( 46, 99)
( 47, 98)( 48,100)( 49, 93)( 50, 95)( 51, 94)( 52, 96)( 53, 77)( 54, 79)
( 55, 78)( 56, 80)( 57, 85)( 58, 87)( 59, 86)( 60, 88)( 61, 81)( 62, 83)
( 63, 82)( 64, 84)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)
( 71,102)( 72,104)( 73,109)( 74,111)( 75,110)( 76,112)(114,115)(117,121)
(118,123)(119,122)(120,124)(125,141)(126,143)(127,142)(128,144)(129,137)
(130,139)(131,138)(132,140)(133,145)(134,147)(135,146)(136,148)(149,197)
(150,199)(151,198)(152,200)(153,205)(154,207)(155,206)(156,208)(157,201)
(158,203)(159,202)(160,204)(161,185)(162,187)(163,186)(164,188)(165,193)
(166,195)(167,194)(168,196)(169,189)(170,191)(171,190)(172,192)(173,213)
(174,215)(175,214)(176,216)(177,209)(178,211)(179,210)(180,212)(181,217)
(182,219)(183,218)(184,220);
s4 := Sym(220)!( 5, 77)( 6, 80)( 7, 79)( 8, 78)( 9, 85)( 10, 88)( 11, 87)
( 12, 86)( 13, 81)( 14, 84)( 15, 83)( 16, 82)( 17,105)( 18,108)( 19,107)
( 20,106)( 21,101)( 22,104)( 23,103)( 24,102)( 25,109)( 26,112)( 27,111)
( 28,110)( 29, 93)( 30, 96)( 31, 95)( 32, 94)( 33, 89)( 34, 92)( 35, 91)
( 36, 90)( 37, 97)( 38,100)( 39, 99)( 40, 98)( 42, 44)( 45, 49)( 46, 52)
( 47, 51)( 48, 50)( 53, 69)( 54, 72)( 55, 71)( 56, 70)( 57, 65)( 58, 68)
( 59, 67)( 60, 66)( 61, 73)( 62, 76)( 63, 75)( 64, 74)(113,185)(114,188)
(115,187)(116,186)(117,193)(118,196)(119,195)(120,194)(121,189)(122,192)
(123,191)(124,190)(125,213)(126,216)(127,215)(128,214)(129,209)(130,212)
(131,211)(132,210)(133,217)(134,220)(135,219)(136,218)(137,201)(138,204)
(139,203)(140,202)(141,197)(142,200)(143,199)(144,198)(145,205)(146,208)
(147,207)(148,206)(150,152)(153,157)(154,160)(155,159)(156,158)(161,177)
(162,180)(163,179)(164,178)(165,173)(166,176)(167,175)(168,174)(169,181)
(170,184)(171,183)(172,182);
poly := sub<Sym(220)|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,
s1*s2*s1*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*s2*s3, s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope