Polytope of Type {6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*864
if this polytope has a name.
Group : SmallGroup(864,3998)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 24, 216, 72
Order of s₀s₁s₂ : 36
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,18,2} of size 1728
Vertex Figure Of :
   {2,6,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,9}*432
   3-fold quotients : {6,6}*288a
   4-fold quotients : {6,18}*216b
   6-fold quotients : {6,3}*144
   8-fold quotients : {6,9}*108
   9-fold quotients : {6,6}*96
   12-fold quotients : {2,18}*72, {6,6}*72b
   18-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {2,9}*36, {6,3}*36
   36-fold quotients : {3,3}*24, {2,6}*24
   72-fold quotients : {2,3}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,36}*1728a, {12,18}*1728a, {6,18}*1728a, {6,36}*1728c, {12,18}*1728b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope