Polytope of Type {2,4,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,27}*864
if this polytope has a name.
Group : SmallGroup(864,1916)
Rank : 4
Schlafli Type : {2,4,27}
Number of vertices, edges, etc : 2, 8, 108, 54
Order of s₀s₁s₂s₃ : 54
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,27,2} of size 1728
Vertex Figure Of :
   {2,2,4,27} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,27}*432
   3-fold quotients : {2,4,9}*288
   4-fold quotients : {2,2,27}*216
   6-fold quotients : {2,4,9}*144
   9-fold quotients : {2,4,3}*96
   12-fold quotients : {2,2,9}*72
   18-fold quotients : {2,4,3}*48
   36-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4,27}*1728b, {2,8,27}*1728, {2,4,54}*1728
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope