Polytope of Type {24,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728w
if this polytope has a name.
Group : SmallGroup(1728,33616)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {12,12}*864l
   3-fold quotients : {24,4}*576b
   4-fold quotients : {6,12}*432i
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,12}*216c
   9-fold quotients : {8,12}*192b
   12-fold quotients : {6,4}*144
   18-fold quotients : {4,12}*96a
   24-fold quotients : {6,4}*72
   27-fold quotients : {8,4}*64b
   36-fold quotients : {2,12}*48, {4,6}*48a
   54-fold quotients : {4,4}*32
   72-fold quotients : {2,6}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {2,3}*12
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope