Polytope of Type {24,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728k
if this polytope has a name.
Group : SmallGroup(1728,12713)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,12}*864f
   3-fold quotients : {24,4}*576b
   4-fold quotients : {6,12}*432f
   6-fold quotients : {12,4}*288
   8-fold quotients : {6,12}*216b
   12-fold quotients : {6,4}*144
   24-fold quotients : {6,4}*72
   27-fold quotients : {8,4}*64b
   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)( 55,163)( 56,164)
( 57,165)( 58,171)( 59,169)( 60,170)( 61,167)( 62,168)( 63,166)( 64,183)
( 65,181)( 66,182)( 67,188)( 68,189)( 69,187)( 70,184)( 71,185)( 72,186)
( 73,173)( 74,174)( 75,172)( 76,178)( 77,179)( 78,180)( 79,177)( 80,175)
( 81,176)( 82,190)( 83,191)( 84,192)( 85,198)( 86,196)( 87,197)( 88,194)
( 89,195)( 90,193)( 91,210)( 92,208)( 93,209)( 94,215)( 95,216)( 96,214)
( 97,211)( 98,212)( 99,213)(100,200)(101,201)(102,199)(103,205)(104,206)
(105,207)(106,204)(107,202)(108,203);;
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, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)( 62, 90)
( 63, 89)( 64, 99)( 65, 98)( 66, 97)( 67, 96)( 68, 95)( 69, 94)( 70, 93)
( 71, 92)( 72, 91)( 73,102)( 74,101)( 75,100)( 76,108)( 77,107)( 78,106)
( 79,105)( 80,104)( 81,103)(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 := (  2,  3)(  4, 22)(  5, 24)(  6, 23)(  7, 18)(  8, 17)(  9, 16)( 10, 19)
( 11, 21)( 12, 20)( 14, 15)( 25, 27)( 29, 30)( 31, 49)( 32, 51)( 33, 50)
( 34, 45)( 35, 44)( 36, 43)( 37, 46)( 38, 48)( 39, 47)( 41, 42)( 52, 54)
( 55, 82)( 56, 84)( 57, 83)( 58,103)( 59,105)( 60,104)( 61, 99)( 62, 98)
( 63, 97)( 64,100)( 65,102)( 66,101)( 67, 94)( 68, 96)( 69, 95)( 70, 90)
( 71, 89)( 72, 88)( 73, 91)( 74, 93)( 75, 92)( 76, 85)( 77, 87)( 78, 86)
( 79,108)( 80,107)( 81,106)(110,111)(112,130)(113,132)(114,131)(115,126)
(116,125)(117,124)(118,127)(119,129)(120,128)(122,123)(133,135)(137,138)
(139,157)(140,159)(141,158)(142,153)(143,152)(144,151)(145,154)(146,156)
(147,155)(149,150)(160,162)(163,190)(164,192)(165,191)(166,211)(167,213)
(168,212)(169,207)(170,206)(171,205)(172,208)(173,210)(174,209)(175,202)
(176,204)(177,203)(178,198)(179,197)(180,196)(181,199)(182,201)(183,200)
(184,193)(185,195)(186,194)(187,216)(188,215)(189,214);;
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*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1, 
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)(  2,137)(  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)( 55,163)
( 56,164)( 57,165)( 58,171)( 59,169)( 60,170)( 61,167)( 62,168)( 63,166)
( 64,183)( 65,181)( 66,182)( 67,188)( 68,189)( 69,187)( 70,184)( 71,185)
( 72,186)( 73,173)( 74,174)( 75,172)( 76,178)( 77,179)( 78,180)( 79,177)
( 80,175)( 81,176)( 82,190)( 83,191)( 84,192)( 85,198)( 86,196)( 87,197)
( 88,194)( 89,195)( 90,193)( 91,210)( 92,208)( 93,209)( 94,215)( 95,216)
( 96,214)( 97,211)( 98,212)( 99,213)(100,200)(101,201)(102,199)(103,205)
(104,206)(105,207)(106,204)(107,202)(108,203);
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, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)
( 62, 90)( 63, 89)( 64, 99)( 65, 98)( 66, 97)( 67, 96)( 68, 95)( 69, 94)
( 70, 93)( 71, 92)( 72, 91)( 73,102)( 74,101)( 75,100)( 76,108)( 77,107)
( 78,106)( 79,105)( 80,104)( 81,103)(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)!(  2,  3)(  4, 22)(  5, 24)(  6, 23)(  7, 18)(  8, 17)(  9, 16)
( 10, 19)( 11, 21)( 12, 20)( 14, 15)( 25, 27)( 29, 30)( 31, 49)( 32, 51)
( 33, 50)( 34, 45)( 35, 44)( 36, 43)( 37, 46)( 38, 48)( 39, 47)( 41, 42)
( 52, 54)( 55, 82)( 56, 84)( 57, 83)( 58,103)( 59,105)( 60,104)( 61, 99)
( 62, 98)( 63, 97)( 64,100)( 65,102)( 66,101)( 67, 94)( 68, 96)( 69, 95)
( 70, 90)( 71, 89)( 72, 88)( 73, 91)( 74, 93)( 75, 92)( 76, 85)( 77, 87)
( 78, 86)( 79,108)( 80,107)( 81,106)(110,111)(112,130)(113,132)(114,131)
(115,126)(116,125)(117,124)(118,127)(119,129)(120,128)(122,123)(133,135)
(137,138)(139,157)(140,159)(141,158)(142,153)(143,152)(144,151)(145,154)
(146,156)(147,155)(149,150)(160,162)(163,190)(164,192)(165,191)(166,211)
(167,213)(168,212)(169,207)(170,206)(171,205)(172,208)(173,210)(174,209)
(175,202)(176,204)(177,203)(178,198)(179,197)(180,196)(181,199)(182,201)
(183,200)(184,193)(185,195)(186,194)(187,216)(188,215)(189,214);
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*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1, 
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