Polytope of Type {6,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4}*864c
if this polytope has a name.
Group : SmallGroup(864,2511)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 18, 54, 36, 4
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,4,2} of size 1728
Vertex Figure Of :
   {2,6,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,4}*432a, {6,6,2}*432c
   3-fold quotients : {6,6,4}*288c
   4-fold quotients : {3,6,2}*216
   6-fold quotients : {3,6,4}*144, {6,6,2}*144c
   9-fold quotients : {6,2,4}*96
   12-fold quotients : {3,6,2}*72
   18-fold quotients : {3,2,4}*48, {6,2,2}*48
   27-fold quotients : {2,2,4}*32
   36-fold quotients : {3,2,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6,4}*1728b, {6,6,8}*1728c, {6,12,4}*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)
( 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)( 55,185)( 56,186)
( 57,184)( 58,183)( 59,181)( 60,182)( 61,187)( 62,188)( 63,189)( 64,176)
( 65,177)( 66,175)( 67,174)( 68,172)( 69,173)( 70,178)( 71,179)( 72,180)
( 73,167)( 74,168)( 75,166)( 76,165)( 77,163)( 78,164)( 79,169)( 80,170)
( 81,171)( 82,212)( 83,213)( 84,211)( 85,210)( 86,208)( 87,209)( 88,214)
( 89,215)( 90,216)( 91,203)( 92,204)( 93,202)( 94,201)( 95,199)( 96,200)
( 97,205)( 98,206)( 99,207)(100,194)(101,195)(102,193)(103,192)(104,190)
(105,191)(106,196)(107,197)(108,198);;
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)(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)(163,190)(164,192)(165,191)(166,196)(167,198)
(168,197)(169,193)(170,195)(171,194)(172,199)(173,201)(174,200)(175,205)
(176,207)(177,206)(178,202)(179,204)(180,203)(181,208)(182,210)(183,209)
(184,214)(185,216)(186,215)(187,211)(188,213)(189,212);;
s3 := (  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)
(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)
( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)
( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)
( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)
( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)
( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)
(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)
(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)
(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)
(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)
(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)
(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)
(159,213)(160,214)(161,215)(162,216);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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)( 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)( 55,185)
( 56,186)( 57,184)( 58,183)( 59,181)( 60,182)( 61,187)( 62,188)( 63,189)
( 64,176)( 65,177)( 66,175)( 67,174)( 68,172)( 69,173)( 70,178)( 71,179)
( 72,180)( 73,167)( 74,168)( 75,166)( 76,165)( 77,163)( 78,164)( 79,169)
( 80,170)( 81,171)( 82,212)( 83,213)( 84,211)( 85,210)( 86,208)( 87,209)
( 88,214)( 89,215)( 90,216)( 91,203)( 92,204)( 93,202)( 94,201)( 95,199)
( 96,200)( 97,205)( 98,206)( 99,207)(100,194)(101,195)(102,193)(103,192)
(104,190)(105,191)(106,196)(107,197)(108,198);
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)(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)(163,190)(164,192)(165,191)(166,196)
(167,198)(168,197)(169,193)(170,195)(171,194)(172,199)(173,201)(174,200)
(175,205)(176,207)(177,206)(178,202)(179,204)(180,203)(181,208)(182,210)
(183,209)(184,214)(185,216)(186,215)(187,211)(188,213)(189,212);
s3 := Sym(216)!(  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)
(  8, 62)(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)
( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)
( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)
( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)
( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)
( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)
(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)
(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)
(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)
(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)
(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)
(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)
(158,212)(159,213)(160,214)(161,215)(162,216);
poly := sub<Sym(216)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope