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

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

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