Polytope of Type {4,4,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,54}*1728
Also Known As : {{4,4|2},{4,54|2}}. if this polytope has another name.
Group : SmallGroup(1728,2390)
Rank : 4
Schlafli Type : {4,4,54}
Number of vertices, edges, etc : 4, 8, 108, 54
Order of s0s1s2s3 : 108
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 : {2,4,54}*864a, {4,2,54}*864
3-fold quotients : {4,4,18}*576
4-fold quotients : {4,2,27}*432, {2,2,54}*432
6-fold quotients : {2,4,18}*288a, {4,2,18}*288
8-fold quotients : {2,2,27}*216
9-fold quotients : {4,4,6}*192
12-fold quotients : {4,2,9}*144, {2,2,18}*144
18-fold quotients : {2,4,6}*96a, {4,2,6}*96
24-fold quotients : {2,2,9}*72
27-fold quotients : {4,4,2}*64
36-fold quotients : {4,2,3}*48, {2,2,6}*48
54-fold quotients : {2,4,2}*32, {4,2,2}*32
72-fold quotients : {2,2,3}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)
(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)
(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)
(133,160)(134,161)(135,162)(163,190)(164,191)(165,192)(166,193)(167,194)
(168,195)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)(175,202)
(176,203)(177,204)(178,205)(179,206)(180,207)(181,208)(182,209)(183,210)
(184,211)(185,212)(186,213)(187,214)(188,215)(189,216);;
s1 := ( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)
( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,163)( 56,164)
( 57,165)( 58,166)( 59,167)( 60,168)( 61,169)( 62,170)( 63,171)( 64,172)
( 65,173)( 66,174)( 67,175)( 68,176)( 69,177)( 70,178)( 71,179)( 72,180)
( 73,181)( 74,182)( 75,183)( 76,184)( 77,185)( 78,186)( 79,187)( 80,188)
( 81,189)( 82,190)( 83,191)( 84,192)( 85,193)( 86,194)( 87,195)( 88,196)
( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)( 94,202)( 95,203)( 96,204)
( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)(104,212)
(105,213)(106,214)(107,215)(108,216);;
s2 := ( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)( 32, 35)
( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)
( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)( 65, 80)
( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 83, 84)
( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)( 95,104)
( 96,103)( 97,102)( 98,101)( 99,100)(109,163)(110,165)(111,164)(112,171)
(113,170)(114,169)(115,168)(116,167)(117,166)(118,189)(119,188)(120,187)
(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,180)(128,179)
(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)(136,190)
(137,192)(138,191)(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)
(145,216)(146,215)(147,214)(148,213)(149,212)(150,211)(151,210)(152,209)
(153,208)(154,207)(155,206)(156,205)(157,204)(158,203)(159,202)(160,201)
(161,200)(162,199);;
s3 := ( 1, 10)( 2, 12)( 3, 11)( 4, 18)( 5, 17)( 6, 16)( 7, 15)( 8, 14)
( 9, 13)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 28, 37)( 29, 39)( 30, 38)
( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 46, 54)( 47, 53)
( 48, 52)( 49, 51)( 55, 64)( 56, 66)( 57, 65)( 58, 72)( 59, 71)( 60, 70)
( 61, 69)( 62, 68)( 63, 67)( 73, 81)( 74, 80)( 75, 79)( 76, 78)( 82, 91)
( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)( 90, 94)
(100,108)(101,107)(102,106)(103,105)(109,118)(110,120)(111,119)(112,126)
(113,125)(114,124)(115,123)(116,122)(117,121)(127,135)(128,134)(129,133)
(130,132)(136,145)(137,147)(138,146)(139,153)(140,152)(141,151)(142,150)
(143,149)(144,148)(154,162)(155,161)(156,160)(157,159)(163,172)(164,174)
(165,173)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(181,189)
(182,188)(183,187)(184,186)(190,199)(191,201)(192,200)(193,207)(194,206)
(195,205)(196,204)(197,203)(198,202)(208,216)(209,215)(210,214)(211,213);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, 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*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)!(109,136)(110,137)(111,138)(112,139)(113,140)(114,141)(115,142)
(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)
(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)
(132,159)(133,160)(134,161)(135,162)(163,190)(164,191)(165,192)(166,193)
(167,194)(168,195)(169,196)(170,197)(171,198)(172,199)(173,200)(174,201)
(175,202)(176,203)(177,204)(178,205)(179,206)(180,207)(181,208)(182,209)
(183,210)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216);
s1 := Sym(216)!( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)
( 8,116)( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)
( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)
( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,163)
( 56,164)( 57,165)( 58,166)( 59,167)( 60,168)( 61,169)( 62,170)( 63,171)
( 64,172)( 65,173)( 66,174)( 67,175)( 68,176)( 69,177)( 70,178)( 71,179)
( 72,180)( 73,181)( 74,182)( 75,183)( 76,184)( 77,185)( 78,186)( 79,187)
( 80,188)( 81,189)( 82,190)( 83,191)( 84,192)( 85,193)( 86,194)( 87,195)
( 88,196)( 89,197)( 90,198)( 91,199)( 92,200)( 93,201)( 94,202)( 95,203)
( 96,204)( 97,205)( 98,206)( 99,207)(100,208)(101,209)(102,210)(103,211)
(104,212)(105,213)(106,214)(107,215)(108,216);
s2 := Sym(216)!( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)
( 32, 35)( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)
( 43, 48)( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)
( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(109,163)(110,165)(111,164)
(112,171)(113,170)(114,169)(115,168)(116,167)(117,166)(118,189)(119,188)
(120,187)(121,186)(122,185)(123,184)(124,183)(125,182)(126,181)(127,180)
(128,179)(129,178)(130,177)(131,176)(132,175)(133,174)(134,173)(135,172)
(136,190)(137,192)(138,191)(139,198)(140,197)(141,196)(142,195)(143,194)
(144,193)(145,216)(146,215)(147,214)(148,213)(149,212)(150,211)(151,210)
(152,209)(153,208)(154,207)(155,206)(156,205)(157,204)(158,203)(159,202)
(160,201)(161,200)(162,199);
s3 := Sym(216)!( 1, 10)( 2, 12)( 3, 11)( 4, 18)( 5, 17)( 6, 16)( 7, 15)
( 8, 14)( 9, 13)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 28, 37)( 29, 39)
( 30, 38)( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 46, 54)
( 47, 53)( 48, 52)( 49, 51)( 55, 64)( 56, 66)( 57, 65)( 58, 72)( 59, 71)
( 60, 70)( 61, 69)( 62, 68)( 63, 67)( 73, 81)( 74, 80)( 75, 79)( 76, 78)
( 82, 91)( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)
( 90, 94)(100,108)(101,107)(102,106)(103,105)(109,118)(110,120)(111,119)
(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(127,135)(128,134)
(129,133)(130,132)(136,145)(137,147)(138,146)(139,153)(140,152)(141,151)
(142,150)(143,149)(144,148)(154,162)(155,161)(156,160)(157,159)(163,172)
(164,174)(165,173)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)
(181,189)(182,188)(183,187)(184,186)(190,199)(191,201)(192,200)(193,207)
(194,206)(195,205)(196,204)(197,203)(198,202)(208,216)(209,215)(210,214)
(211,213);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
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*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