Polytope of Type {6,18,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18,4}*864d
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 4
Schlafli Type : {6,18,4}
Number of vertices, edges, etc : 6, 54, 36, 4
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,18,4,2} of size 1728
Vertex Figure Of :
   {2,6,18,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,9,4}*432
   3-fold quotients : {2,18,4}*288b, {6,6,4}*288e
   6-fold quotients : {2,9,4}*144, {6,3,4}*144
   9-fold quotients : {2,6,4}*96c
   18-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,36,4}*1728e, {6,36,4}*1728f, {12,18,4}*1728d, {6,18,4}*1728b
Permutation Representation (GAP) :

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