Polytope of Type {6,24}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24}*864c
if this polytope has a name.
Group : SmallGroup(864,1157)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 18, 216, 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 :
   {6,24,2} of size 1728
Vertex Figure Of :
   {2,6,24} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,12}*432c
   3-fold quotients : {6,24}*288c
   4-fold quotients : {6,6}*216c
   6-fold quotients : {6,12}*144c
   8-fold quotients : {3,6}*108
   9-fold quotients : {6,8}*96
   12-fold quotients : {6,6}*72c
   18-fold quotients : {6,4}*48a
   24-fold quotients : {3,6}*36
   27-fold quotients : {2,8}*32
   36-fold quotients : {6,2}*24
   54-fold quotients : {2,4}*16
   72-fold quotients : {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,24}*1728b, {6,48}*1728c
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      24 facets:
         24 of {6}*12
      10 vertex figures:
         4 of {24}*48
         6 of {8}*16

Permutation Representation (GAP) : 
s0 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105)(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)(145,154)(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)(153,159)(166,169)(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)(200,209)(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)(207,213);;
s1 := (  1,131)(  2,132)(  3,130)(  4,129)(  5,127)(  6,128)(  7,133)(  8,134)(  9,135)( 10,122)( 11,123)( 12,121)( 13,120)( 14,118)( 15,119)( 16,124)( 17,125)( 18,126)( 19,113)( 20,114)( 21,112)( 22,111)( 23,109)( 24,110)( 25,115)( 26,116)( 27,117)( 28,158)( 29,159)( 30,157)( 31,156)( 32,154)( 33,155)( 34,160)( 35,161)( 36,162)( 37,149)( 38,150)( 39,148)( 40,147)( 41,145)( 42,146)( 43,151)( 44,152)( 45,153)( 46,140)( 47,141)( 48,139)( 49,138)( 50,136)( 51,137)( 52,142)( 53,143)( 54,144)( 55,212)( 56,213)( 57,211)( 58,210)( 59,208)( 60,209)( 61,214)( 62,215)( 63,216)( 64,203)( 65,204)( 66,202)( 67,201)( 68,199)( 69,200)( 70,205)( 71,206)( 72,207)( 73,194)( 74,195)( 75,193)( 76,192)( 77,190)( 78,191)( 79,196)( 80,197)( 81,198)( 82,185)( 83,186)( 84,184)( 85,183)( 86,181)( 87,182)( 88,187)( 89,188)( 90,189)( 91,176)( 92,177)( 93,175)( 94,174)( 95,172)( 96,173)( 97,178)( 98,179)( 99,180)(100,167)(101,168)(102,166)(103,165)(104,163)(105,164)(106,169)(107,170)(108,171);;
s2 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)( 70, 94)( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)( 78,107)( 79,103)( 80,105)( 81,104)(109,163)(110,165)(111,164)(112,169)(113,171)(114,170)(115,166)(116,168)(117,167)(118,172)(119,174)(120,173)(121,178)(122,180)(123,179)(124,175)(125,177)(126,176)(127,181)(128,183)(129,182)(130,187)(131,189)(132,188)(133,184)(134,186)(135,185)(136,190)(137,192)(138,191)(139,196)(140,198)(141,197)(142,193)(143,195)(144,194)(145,199)(146,201)(147,200)(148,205)(149,207)(150,206)(151,202)(152,204)(153,203)(154,208)(155,210)(156,209)(157,214)(158,216)(159,215)(160,211)(161,213)(162,212);;
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*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s1*s2*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(216)!(  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105)(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)(145,154)(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)(153,159)(166,169)(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)(200,209)(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)(207,213);
s1 := Sym(216)!(  1,131)(  2,132)(  3,130)(  4,129)(  5,127)(  6,128)(  7,133)(  8,134)(  9,135)( 10,122)( 11,123)( 12,121)( 13,120)( 14,118)( 15,119)( 16,124)( 17,125)( 18,126)( 19,113)( 20,114)( 21,112)( 22,111)( 23,109)( 24,110)( 25,115)( 26,116)( 27,117)( 28,158)( 29,159)( 30,157)( 31,156)( 32,154)( 33,155)( 34,160)( 35,161)( 36,162)( 37,149)( 38,150)( 39,148)( 40,147)( 41,145)( 42,146)( 43,151)( 44,152)( 45,153)( 46,140)( 47,141)( 48,139)( 49,138)( 50,136)( 51,137)( 52,142)( 53,143)( 54,144)( 55,212)( 56,213)( 57,211)( 58,210)( 59,208)( 60,209)( 61,214)( 62,215)( 63,216)( 64,203)( 65,204)( 66,202)( 67,201)( 68,199)( 69,200)( 70,205)( 71,206)( 72,207)( 73,194)( 74,195)( 75,193)( 76,192)( 77,190)( 78,191)( 79,196)( 80,197)( 81,198)( 82,185)( 83,186)( 84,184)( 85,183)( 86,181)( 87,182)( 88,187)( 89,188)( 90,189)( 91,176)( 92,177)( 93,175)( 94,174)( 95,172)( 96,173)( 97,178)( 98,179)( 99,180)(100,167)(101,168)(102,166)(103,165)(104,163)(105,164)(106,169)(107,170)(108,171);
s2 := Sym(216)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)( 70, 94)( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)( 78,107)( 79,103)( 80,105)( 81,104)(109,163)(110,165)(111,164)(112,169)(113,171)(114,170)(115,166)(116,168)(117,167)(118,172)(119,174)(120,173)(121,178)(122,180)(123,179)(124,175)(125,177)(126,176)(127,181)(128,183)(129,182)(130,187)(131,189)(132,188)(133,184)(134,186)(135,185)(136,190)(137,192)(138,191)(139,196)(140,198)(141,197)(142,193)(143,195)(144,194)(145,199)(146,201)(147,200)(148,205)(149,207)(150,206)(151,202)(152,204)(153,203)(154,208)(155,210)(156,209)(157,214)(158,216)(159,215)(160,211)(161,213)(162,212);
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*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s1*s2*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s2 >;
References : None.
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