Polytope of Type {24,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,6}*864a
if this polytope has a name.
Group : SmallGroup(864,814)
Rank : 3
Schlafli Type : {24,6}
Number of vertices, edges, etc : 72, 216, 18
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {24,6,2} of size 1728
Vertex Figure Of :
   {2,24,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,6}*432a
   3-fold quotients : {24,6}*288b
   4-fold quotients : {6,6}*216c
   6-fold quotients : {12,6}*144b
   8-fold quotients : {3,6}*108
   9-fold quotients : {24,2}*96
   12-fold quotients : {6,6}*72c
   18-fold quotients : {12,2}*48
   24-fold quotients : {3,6}*36
   27-fold quotients : {8,2}*32
   36-fold quotients : {6,2}*24
   54-fold quotients : {4,2}*16
   72-fold quotients : {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {48,6}*1728a, {24,12}*1728c
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)
( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)( 62, 86)
( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)( 70,103)
( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)( 78, 99)
( 79, 94)( 80, 95)( 81, 96)(109,163)(110,164)(111,165)(112,169)(113,170)
(114,171)(115,166)(116,167)(117,168)(118,181)(119,182)(120,183)(121,187)
(122,188)(123,189)(124,184)(125,185)(126,186)(127,172)(128,173)(129,174)
(130,178)(131,179)(132,180)(133,175)(134,176)(135,177)(136,190)(137,191)
(138,192)(139,196)(140,197)(141,198)(142,193)(143,194)(144,195)(145,208)
(146,209)(147,210)(148,214)(149,215)(150,216)(151,211)(152,212)(153,213)
(154,199)(155,200)(156,201)(157,205)(158,206)(159,207)(160,202)(161,203)
(162,204);;
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)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)
(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)
(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)
(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)
(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)
(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)
(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)(213,215);;
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*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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*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)!(  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)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)
( 62, 86)( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)
( 70,103)( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)
( 78, 99)( 79, 94)( 80, 95)( 81, 96)(109,163)(110,164)(111,165)(112,169)
(113,170)(114,171)(115,166)(116,167)(117,168)(118,181)(119,182)(120,183)
(121,187)(122,188)(123,189)(124,184)(125,185)(126,186)(127,172)(128,173)
(129,174)(130,178)(131,179)(132,180)(133,175)(134,176)(135,177)(136,190)
(137,191)(138,192)(139,196)(140,197)(141,198)(142,193)(143,194)(144,195)
(145,208)(146,209)(147,210)(148,214)(149,215)(150,216)(151,211)(152,212)
(153,213)(154,199)(155,200)(156,201)(157,205)(158,206)(159,207)(160,202)
(161,203)(162,204);
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)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)
(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)
(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)
(159,161)(164,165)(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)
(177,179)(182,183)(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)
(195,197)(200,201)(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)
(213,215);
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*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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*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