Polytope of Type {12,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*1728v
if this polytope has a name.
Group : SmallGroup(1728,33596)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 36, 432, 72
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 6
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}*864k
   3-fold quotients : {4,24}*576a
   4-fold quotients : {12,6}*432i
   6-fold quotients : {4,12}*288
   8-fold quotients : {12,6}*216c
   9-fold quotients : {12,8}*192a
   12-fold quotients : {4,6}*144
   18-fold quotients : {12,4}*96a, {6,8}*96
   24-fold quotients : {4,6}*72
   27-fold quotients : {4,8}*64a
   36-fold quotients : {12,2}*48, {6,4}*48a
   54-fold quotients : {4,4}*32, {2,8}*32
   72-fold quotients : {6,2}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {3,2}*12
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope