Polytope of Type {4,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,27}*432
if this polytope has a name.
Group : SmallGroup(432,224)
Rank : 3
Schlafli Type : {4,27}
Number of vertices, edges, etc : 8, 108, 54
Order of s₀s₁s₂ : 54
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,27,2} of size 864
   {4,27,4} of size 1728
Vertex Figure Of :
   {2,4,27} of size 864
   {4,4,27} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,27}*216
   3-fold quotients : {4,9}*144
   4-fold quotients : {2,27}*108
   6-fold quotients : {4,9}*72
   9-fold quotients : {4,3}*48
   12-fold quotients : {2,9}*36
   18-fold quotients : {4,3}*24
   36-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,27}*864, {4,54}*864
   3-fold covers : {4,81}*1296, {12,27}*1296
   4-fold covers : {8,27}*1728, {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope