Polytope of Type {54,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,4,4}*1728
Also Known As : {{54,4|2},{4,4|2}}. if this polytope has another name.
Group : SmallGroup(1728,2390)
Rank : 4
Schlafli Type : {54,4,4}
Number of vertices, edges, etc : 54, 108, 8, 4
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 : {54,2,4}*864, {54,4,2}*864a
3-fold quotients : {18,4,4}*576
4-fold quotients : {27,2,4}*432, {54,2,2}*432
6-fold quotients : {18,2,4}*288, {18,4,2}*288a
8-fold quotients : {27,2,2}*216
9-fold quotients : {6,4,4}*192
12-fold quotients : {9,2,4}*144, {18,2,2}*144
18-fold quotients : {6,2,4}*96, {6,4,2}*96a
24-fold quotients : {9,2,2}*72
27-fold quotients : {2,4,4}*64
36-fold quotients : {3,2,4}*48, {6,2,2}*48
54-fold quotients : {2,2,4}*32, {2,4,2}*32
72-fold quotients : {3,2,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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)(110,111)(112,117)(113,116)(114,115)
(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)
(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)(147,160)
(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(164,165)(166,171)
(167,170)(168,169)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)
(178,183)(179,182)(180,181)(191,192)(193,198)(194,197)(195,196)(199,216)
(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208);;
s1 := ( 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)(109,145)(110,147)(111,146)(112,153)
(113,152)(114,151)(115,150)(116,149)(117,148)(118,136)(119,138)(120,137)
(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,162)(128,161)
(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(163,199)
(164,201)(165,200)(166,207)(167,206)(168,205)(169,204)(170,203)(171,202)
(172,190)(173,192)(174,191)(175,198)(176,197)(177,196)(178,195)(179,194)
(180,193)(181,216)(182,215)(183,214)(184,213)(185,212)(186,211)(187,210)
(188,209)(189,208);;
s2 := ( 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);;
s3 := (109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)
(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)
(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)
(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)
(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)
(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)
(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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)(110,111)(112,117)(113,116)
(114,115)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)
(125,128)(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)
(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(164,165)
(166,171)(167,170)(168,169)(172,189)(173,188)(174,187)(175,186)(176,185)
(177,184)(178,183)(179,182)(180,181)(191,192)(193,198)(194,197)(195,196)
(199,216)(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)
(207,208);
s1 := 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)(109,145)(110,147)(111,146)
(112,153)(113,152)(114,151)(115,150)(116,149)(117,148)(118,136)(119,138)
(120,137)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)
(163,199)(164,201)(165,200)(166,207)(167,206)(168,205)(169,204)(170,203)
(171,202)(172,190)(173,192)(174,191)(175,198)(176,197)(177,196)(178,195)
(179,194)(180,193)(181,216)(182,215)(183,214)(184,213)(185,212)(186,211)
(187,210)(188,209)(189,208);
s2 := 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);
s3 := Sym(216)!(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)
(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)
(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)
(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)
(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)
(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)
(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);
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, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope