Polytope of Type {12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,12630)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 216, 432, 72
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,4}*864b
   3-fold quotients : {12,4}*576
   4-fold quotients : {6,4}*432a
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,4}*216
   12-fold quotients : {6,4}*144
   24-fold quotients : {6,4}*72
   27-fold quotients : {4,4}*64
   54-fold quotients : {4,4}*32
   108-fold quotients : {2,4}*16, {4,2}*16
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope