Polytope of Type {4,54,2,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,54,2,2}*1728b
if this polytope has a name.
Group : SmallGroup(1728,20782)
Rank : 5
Schlafli Type : {4,54,2,2}
Number of vertices, edges, etc : 4, 108, 54, 2, 2
Order of s₀s₁s₂s₃s₄ : 54
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {4,27,2,2}*864
   3-fold quotients : {4,18,2,2}*576b
   6-fold quotients : {4,9,2,2}*288
   9-fold quotients : {4,6,2,2}*192c
   18-fold quotients : {4,3,2,2}*96
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216);;
s1 := (  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);;
s2 := (  1,181)(  2,184)(  3,183)(  4,182)(  5,189)(  6,192)(  7,191)(  8,190)
(  9,185)( 10,188)( 11,187)( 12,186)( 13,209)( 14,212)( 15,211)( 16,210)
( 17,205)( 18,208)( 19,207)( 20,206)( 21,213)( 22,216)( 23,215)( 24,214)
( 25,197)( 26,200)( 27,199)( 28,198)( 29,193)( 30,196)( 31,195)( 32,194)
( 33,201)( 34,204)( 35,203)( 36,202)( 37,145)( 38,148)( 39,147)( 40,146)
( 41,153)( 42,156)( 43,155)( 44,154)( 45,149)( 46,152)( 47,151)( 48,150)
( 49,173)( 50,176)( 51,175)( 52,174)( 53,169)( 54,172)( 55,171)( 56,170)
( 57,177)( 58,180)( 59,179)( 60,178)( 61,161)( 62,164)( 63,163)( 64,162)
( 65,157)( 66,160)( 67,159)( 68,158)( 69,165)( 70,168)( 71,167)( 72,166)
( 73,109)( 74,112)( 75,111)( 76,110)( 77,117)( 78,120)( 79,119)( 80,118)
( 81,113)( 82,116)( 83,115)( 84,114)( 85,137)( 86,140)( 87,139)( 88,138)
( 89,133)( 90,136)( 91,135)( 92,134)( 93,141)( 94,144)( 95,143)( 96,142)
( 97,125)( 98,128)( 99,127)(100,126)(101,121)(102,124)(103,123)(104,122)
(105,129)(106,132)(107,131)(108,130);;
s3 := (217,218);;
s4 := (219,220);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s0*s1, 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*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*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*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*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(220)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216);
s1 := Sym(220)!(  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);
s2 := Sym(220)!(  1,181)(  2,184)(  3,183)(  4,182)(  5,189)(  6,192)(  7,191)
(  8,190)(  9,185)( 10,188)( 11,187)( 12,186)( 13,209)( 14,212)( 15,211)
( 16,210)( 17,205)( 18,208)( 19,207)( 20,206)( 21,213)( 22,216)( 23,215)
( 24,214)( 25,197)( 26,200)( 27,199)( 28,198)( 29,193)( 30,196)( 31,195)
( 32,194)( 33,201)( 34,204)( 35,203)( 36,202)( 37,145)( 38,148)( 39,147)
( 40,146)( 41,153)( 42,156)( 43,155)( 44,154)( 45,149)( 46,152)( 47,151)
( 48,150)( 49,173)( 50,176)( 51,175)( 52,174)( 53,169)( 54,172)( 55,171)
( 56,170)( 57,177)( 58,180)( 59,179)( 60,178)( 61,161)( 62,164)( 63,163)
( 64,162)( 65,157)( 66,160)( 67,159)( 68,158)( 69,165)( 70,168)( 71,167)
( 72,166)( 73,109)( 74,112)( 75,111)( 76,110)( 77,117)( 78,120)( 79,119)
( 80,118)( 81,113)( 82,116)( 83,115)( 84,114)( 85,137)( 86,140)( 87,139)
( 88,138)( 89,133)( 90,136)( 91,135)( 92,134)( 93,141)( 94,144)( 95,143)
( 96,142)( 97,125)( 98,128)( 99,127)(100,126)(101,121)(102,124)(103,123)
(104,122)(105,129)(106,132)(107,131)(108,130);
s3 := Sym(220)!(217,218);
s4 := Sym(220)!(219,220);
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*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s0*s1, 
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*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*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*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*s1*s2*s1*s2 >;

to this polytope