Polytope of Type {54,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,4}*432c
if this polytope has a name.
Group : SmallGroup(432,224)
Rank : 3
Schlafli Type : {54,4}
Number of vertices, edges, etc : 54, 108, 4
Order of s0s1s2 : 27
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{54,4,2} of size 864
Vertex Figure Of :
{2,54,4} of size 864
{4,54,4} of size 1728
{4,54,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {27,4}*216
3-fold quotients : {18,4}*144c
6-fold quotients : {9,4}*72
9-fold quotients : {6,4}*48b
18-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {54,4}*864
3-fold covers : {162,4}*1296c, {54,12}*1296c
4-fold covers : {54,8}*1728a, {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 93)( 42, 95)( 43, 94)
( 44, 96)( 45, 89)( 46, 91)( 47, 90)( 48, 92)( 49, 73)( 50, 75)( 51, 74)
( 52, 76)( 53, 81)( 54, 83)( 55, 82)( 56, 84)( 57, 77)( 58, 79)( 59, 78)
( 60, 80)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)( 67, 98)
( 68,100)( 69,105)( 70,107)( 71,106)( 72,108)(110,111)(113,117)(114,119)
(115,118)(116,120)(121,137)(122,139)(123,138)(124,140)(125,133)(126,135)
(127,134)(128,136)(129,141)(130,143)(131,142)(132,144)(145,193)(146,195)
(147,194)(148,196)(149,201)(150,203)(151,202)(152,204)(153,197)(154,199)
(155,198)(156,200)(157,181)(158,183)(159,182)(160,184)(161,189)(162,191)
(163,190)(164,192)(165,185)(166,187)(167,186)(168,188)(169,209)(170,211)
(171,210)(172,212)(173,205)(174,207)(175,206)(176,208)(177,213)(178,215)
(179,214)(180,216);;
s1 := ( 1,145)( 2,146)( 3,148)( 4,147)( 5,153)( 6,154)( 7,156)( 8,155)
( 9,149)( 10,150)( 11,152)( 12,151)( 13,173)( 14,174)( 15,176)( 16,175)
( 17,169)( 18,170)( 19,172)( 20,171)( 21,177)( 22,178)( 23,180)( 24,179)
( 25,161)( 26,162)( 27,164)( 28,163)( 29,157)( 30,158)( 31,160)( 32,159)
( 33,165)( 34,166)( 35,168)( 36,167)( 37,109)( 38,110)( 39,112)( 40,111)
( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)( 48,115)
( 49,137)( 50,138)( 51,140)( 52,139)( 53,133)( 54,134)( 55,136)( 56,135)
( 57,141)( 58,142)( 59,144)( 60,143)( 61,125)( 62,126)( 63,128)( 64,127)
( 65,121)( 66,122)( 67,124)( 68,123)( 69,129)( 70,130)( 71,132)( 72,131)
( 73,193)( 74,194)( 75,196)( 76,195)( 77,201)( 78,202)( 79,204)( 80,203)
( 81,197)( 82,198)( 83,200)( 84,199)( 85,181)( 86,182)( 87,184)( 88,183)
( 89,189)( 90,190)( 91,192)( 92,191)( 93,185)( 94,186)( 95,188)( 96,187)
( 97,209)( 98,210)( 99,212)(100,211)(101,205)(102,206)(103,208)(104,207)
(105,213)(106,214)(107,216)(108,215);;
s2 := ( 1,112)( 2,111)( 3,110)( 4,109)( 5,116)( 6,115)( 7,114)( 8,113)
( 9,120)( 10,119)( 11,118)( 12,117)( 13,124)( 14,123)( 15,122)( 16,121)
( 17,128)( 18,127)( 19,126)( 20,125)( 21,132)( 22,131)( 23,130)( 24,129)
( 25,136)( 26,135)( 27,134)( 28,133)( 29,140)( 30,139)( 31,138)( 32,137)
( 33,144)( 34,143)( 35,142)( 36,141)( 37,148)( 38,147)( 39,146)( 40,145)
( 41,152)( 42,151)( 43,150)( 44,149)( 45,156)( 46,155)( 47,154)( 48,153)
( 49,160)( 50,159)( 51,158)( 52,157)( 53,164)( 54,163)( 55,162)( 56,161)
( 57,168)( 58,167)( 59,166)( 60,165)( 61,172)( 62,171)( 63,170)( 64,169)
( 65,176)( 66,175)( 67,174)( 68,173)( 69,180)( 70,179)( 71,178)( 72,177)
( 73,184)( 74,183)( 75,182)( 76,181)( 77,188)( 78,187)( 79,186)( 80,185)
( 81,192)( 82,191)( 83,190)( 84,189)( 85,196)( 86,195)( 87,194)( 88,193)
( 89,200)( 90,199)( 91,198)( 92,197)( 93,204)( 94,203)( 95,202)( 96,201)
( 97,208)( 98,207)( 99,206)(100,205)(101,212)(102,211)(103,210)(104,209)
(105,216)(106,215)(107,214)(108,213);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s0*s2*s1*s0*s2*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*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 29)( 14, 31)
( 15, 30)( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)
( 23, 34)( 24, 36)( 37, 85)( 38, 87)( 39, 86)( 40, 88)( 41, 93)( 42, 95)
( 43, 94)( 44, 96)( 45, 89)( 46, 91)( 47, 90)( 48, 92)( 49, 73)( 50, 75)
( 51, 74)( 52, 76)( 53, 81)( 54, 83)( 55, 82)( 56, 84)( 57, 77)( 58, 79)
( 59, 78)( 60, 80)( 61,101)( 62,103)( 63,102)( 64,104)( 65, 97)( 66, 99)
( 67, 98)( 68,100)( 69,105)( 70,107)( 71,106)( 72,108)(110,111)(113,117)
(114,119)(115,118)(116,120)(121,137)(122,139)(123,138)(124,140)(125,133)
(126,135)(127,134)(128,136)(129,141)(130,143)(131,142)(132,144)(145,193)
(146,195)(147,194)(148,196)(149,201)(150,203)(151,202)(152,204)(153,197)
(154,199)(155,198)(156,200)(157,181)(158,183)(159,182)(160,184)(161,189)
(162,191)(163,190)(164,192)(165,185)(166,187)(167,186)(168,188)(169,209)
(170,211)(171,210)(172,212)(173,205)(174,207)(175,206)(176,208)(177,213)
(178,215)(179,214)(180,216);
s1 := Sym(216)!( 1,145)( 2,146)( 3,148)( 4,147)( 5,153)( 6,154)( 7,156)
( 8,155)( 9,149)( 10,150)( 11,152)( 12,151)( 13,173)( 14,174)( 15,176)
( 16,175)( 17,169)( 18,170)( 19,172)( 20,171)( 21,177)( 22,178)( 23,180)
( 24,179)( 25,161)( 26,162)( 27,164)( 28,163)( 29,157)( 30,158)( 31,160)
( 32,159)( 33,165)( 34,166)( 35,168)( 36,167)( 37,109)( 38,110)( 39,112)
( 40,111)( 41,117)( 42,118)( 43,120)( 44,119)( 45,113)( 46,114)( 47,116)
( 48,115)( 49,137)( 50,138)( 51,140)( 52,139)( 53,133)( 54,134)( 55,136)
( 56,135)( 57,141)( 58,142)( 59,144)( 60,143)( 61,125)( 62,126)( 63,128)
( 64,127)( 65,121)( 66,122)( 67,124)( 68,123)( 69,129)( 70,130)( 71,132)
( 72,131)( 73,193)( 74,194)( 75,196)( 76,195)( 77,201)( 78,202)( 79,204)
( 80,203)( 81,197)( 82,198)( 83,200)( 84,199)( 85,181)( 86,182)( 87,184)
( 88,183)( 89,189)( 90,190)( 91,192)( 92,191)( 93,185)( 94,186)( 95,188)
( 96,187)( 97,209)( 98,210)( 99,212)(100,211)(101,205)(102,206)(103,208)
(104,207)(105,213)(106,214)(107,216)(108,215);
s2 := Sym(216)!( 1,112)( 2,111)( 3,110)( 4,109)( 5,116)( 6,115)( 7,114)
( 8,113)( 9,120)( 10,119)( 11,118)( 12,117)( 13,124)( 14,123)( 15,122)
( 16,121)( 17,128)( 18,127)( 19,126)( 20,125)( 21,132)( 22,131)( 23,130)
( 24,129)( 25,136)( 26,135)( 27,134)( 28,133)( 29,140)( 30,139)( 31,138)
( 32,137)( 33,144)( 34,143)( 35,142)( 36,141)( 37,148)( 38,147)( 39,146)
( 40,145)( 41,152)( 42,151)( 43,150)( 44,149)( 45,156)( 46,155)( 47,154)
( 48,153)( 49,160)( 50,159)( 51,158)( 52,157)( 53,164)( 54,163)( 55,162)
( 56,161)( 57,168)( 58,167)( 59,166)( 60,165)( 61,172)( 62,171)( 63,170)
( 64,169)( 65,176)( 66,175)( 67,174)( 68,173)( 69,180)( 70,179)( 71,178)
( 72,177)( 73,184)( 74,183)( 75,182)( 76,181)( 77,188)( 78,187)( 79,186)
( 80,185)( 81,192)( 82,191)( 83,190)( 84,189)( 85,196)( 86,195)( 87,194)
( 88,193)( 89,200)( 90,199)( 91,198)( 92,197)( 93,204)( 94,203)( 95,202)
( 96,201)( 97,208)( 98,207)( 99,206)(100,205)(101,212)(102,211)(103,210)
(104,209)(105,216)(106,215)(107,214)(108,213);
poly := sub<Sym(216)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s0*s2*s1*s0*s2*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*s2 >;
References : None.
to this polytope