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Polytope of Type {24,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,6,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,33799)
Rank : 4
Schlafli Type : {24,6,6}
Number of vertices, edges, etc : 24, 72, 18, 6
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,6,6}*864c
   3-fold quotients : {24,6,2}*576a, {8,6,6}*576b
   4-fold quotients : {6,6,6}*432e
   6-fold quotients : {12,6,2}*288a, {4,6,6}*288b
   9-fold quotients : {24,2,2}*192, {8,6,2}*192
   12-fold quotients : {2,6,6}*144c, {6,6,2}*144a
   18-fold quotients : {12,2,2}*96, {4,6,2}*96a
   24-fold quotients : {2,3,6}*72
   27-fold quotients : {8,2,2}*64
   36-fold quotients : {2,6,2}*48, {6,2,2}*48
   54-fold quotients : {4,2,2}*32
   72-fold quotients : {2,3,2}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)
( 50, 51)( 53, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)
( 61, 88)( 62, 90)( 63, 89)( 64, 91)( 65, 93)( 66, 92)( 67, 94)( 68, 96)
( 69, 95)( 70, 97)( 71, 99)( 72, 98)( 73,100)( 74,102)( 75,101)( 76,103)
( 77,105)( 78,104)( 79,106)( 80,108)( 81,107)(109,163)(110,165)(111,164)
(112,166)(113,168)(114,167)(115,169)(116,171)(117,170)(118,172)(119,174)
(120,173)(121,175)(122,177)(123,176)(124,178)(125,180)(126,179)(127,181)
(128,183)(129,182)(130,184)(131,186)(132,185)(133,187)(134,189)(135,188)
(136,190)(137,192)(138,191)(139,193)(140,195)(141,194)(142,196)(143,198)
(144,197)(145,199)(146,201)(147,200)(148,202)(149,204)(150,203)(151,205)
(152,207)(153,206)(154,208)(155,210)(156,209)(157,211)(158,213)(159,212)
(160,214)(161,216)(162,215);;
s1 := (  1,110)(  2,109)(  3,111)(  4,116)(  5,115)(  6,117)(  7,113)(  8,112)
(  9,114)( 10,128)( 11,127)( 12,129)( 13,134)( 14,133)( 15,135)( 16,131)
( 17,130)( 18,132)( 19,119)( 20,118)( 21,120)( 22,125)( 23,124)( 24,126)
( 25,122)( 26,121)( 27,123)( 28,137)( 29,136)( 30,138)( 31,143)( 32,142)
( 33,144)( 34,140)( 35,139)( 36,141)( 37,155)( 38,154)( 39,156)( 40,161)
( 41,160)( 42,162)( 43,158)( 44,157)( 45,159)( 46,146)( 47,145)( 48,147)
( 49,152)( 50,151)( 51,153)( 52,149)( 53,148)( 54,150)( 55,191)( 56,190)
( 57,192)( 58,197)( 59,196)( 60,198)( 61,194)( 62,193)( 63,195)( 64,209)
( 65,208)( 66,210)( 67,215)( 68,214)( 69,216)( 70,212)( 71,211)( 72,213)
( 73,200)( 74,199)( 75,201)( 76,206)( 77,205)( 78,207)( 79,203)( 80,202)
( 81,204)( 82,164)( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)( 88,167)
( 89,166)( 90,168)( 91,182)( 92,181)( 93,183)( 94,188)( 95,187)( 96,189)
( 97,185)( 98,184)( 99,186)(100,173)(101,172)(102,174)(103,179)(104,178)
(105,180)(106,176)(107,175)(108,177);;
s2 := (  1, 13)(  2, 14)(  3, 15)(  4, 10)(  5, 11)(  6, 12)(  7, 16)(  8, 17)
(  9, 18)( 19, 22)( 20, 23)( 21, 24)( 28, 40)( 29, 41)( 30, 42)( 31, 37)
( 32, 38)( 33, 39)( 34, 43)( 35, 44)( 36, 45)( 46, 49)( 47, 50)( 48, 51)
( 55, 67)( 56, 68)( 57, 69)( 58, 64)( 59, 65)( 60, 66)( 61, 70)( 62, 71)
( 63, 72)( 73, 76)( 74, 77)( 75, 78)( 82, 94)( 83, 95)( 84, 96)( 85, 91)
( 86, 92)( 87, 93)( 88, 97)( 89, 98)( 90, 99)(100,103)(101,104)(102,105)
(109,121)(110,122)(111,123)(112,118)(113,119)(114,120)(115,124)(116,125)
(117,126)(127,130)(128,131)(129,132)(136,148)(137,149)(138,150)(139,145)
(140,146)(141,147)(142,151)(143,152)(144,153)(154,157)(155,158)(156,159)
(163,175)(164,176)(165,177)(166,172)(167,173)(168,174)(169,178)(170,179)
(171,180)(181,184)(182,185)(183,186)(190,202)(191,203)(192,204)(193,199)
(194,200)(195,201)(196,205)(197,206)(198,207)(208,211)(209,212)(210,213);;
s3 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)
(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)
(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)
(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
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)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)
( 47, 48)( 50, 51)( 53, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)
( 60, 86)( 61, 88)( 62, 90)( 63, 89)( 64, 91)( 65, 93)( 66, 92)( 67, 94)
( 68, 96)( 69, 95)( 70, 97)( 71, 99)( 72, 98)( 73,100)( 74,102)( 75,101)
( 76,103)( 77,105)( 78,104)( 79,106)( 80,108)( 81,107)(109,163)(110,165)
(111,164)(112,166)(113,168)(114,167)(115,169)(116,171)(117,170)(118,172)
(119,174)(120,173)(121,175)(122,177)(123,176)(124,178)(125,180)(126,179)
(127,181)(128,183)(129,182)(130,184)(131,186)(132,185)(133,187)(134,189)
(135,188)(136,190)(137,192)(138,191)(139,193)(140,195)(141,194)(142,196)
(143,198)(144,197)(145,199)(146,201)(147,200)(148,202)(149,204)(150,203)
(151,205)(152,207)(153,206)(154,208)(155,210)(156,209)(157,211)(158,213)
(159,212)(160,214)(161,216)(162,215);
s1 := Sym(216)!(  1,110)(  2,109)(  3,111)(  4,116)(  5,115)(  6,117)(  7,113)
(  8,112)(  9,114)( 10,128)( 11,127)( 12,129)( 13,134)( 14,133)( 15,135)
( 16,131)( 17,130)( 18,132)( 19,119)( 20,118)( 21,120)( 22,125)( 23,124)
( 24,126)( 25,122)( 26,121)( 27,123)( 28,137)( 29,136)( 30,138)( 31,143)
( 32,142)( 33,144)( 34,140)( 35,139)( 36,141)( 37,155)( 38,154)( 39,156)
( 40,161)( 41,160)( 42,162)( 43,158)( 44,157)( 45,159)( 46,146)( 47,145)
( 48,147)( 49,152)( 50,151)( 51,153)( 52,149)( 53,148)( 54,150)( 55,191)
( 56,190)( 57,192)( 58,197)( 59,196)( 60,198)( 61,194)( 62,193)( 63,195)
( 64,209)( 65,208)( 66,210)( 67,215)( 68,214)( 69,216)( 70,212)( 71,211)
( 72,213)( 73,200)( 74,199)( 75,201)( 76,206)( 77,205)( 78,207)( 79,203)
( 80,202)( 81,204)( 82,164)( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)
( 88,167)( 89,166)( 90,168)( 91,182)( 92,181)( 93,183)( 94,188)( 95,187)
( 96,189)( 97,185)( 98,184)( 99,186)(100,173)(101,172)(102,174)(103,179)
(104,178)(105,180)(106,176)(107,175)(108,177);
s2 := Sym(216)!(  1, 13)(  2, 14)(  3, 15)(  4, 10)(  5, 11)(  6, 12)(  7, 16)
(  8, 17)(  9, 18)( 19, 22)( 20, 23)( 21, 24)( 28, 40)( 29, 41)( 30, 42)
( 31, 37)( 32, 38)( 33, 39)( 34, 43)( 35, 44)( 36, 45)( 46, 49)( 47, 50)
( 48, 51)( 55, 67)( 56, 68)( 57, 69)( 58, 64)( 59, 65)( 60, 66)( 61, 70)
( 62, 71)( 63, 72)( 73, 76)( 74, 77)( 75, 78)( 82, 94)( 83, 95)( 84, 96)
( 85, 91)( 86, 92)( 87, 93)( 88, 97)( 89, 98)( 90, 99)(100,103)(101,104)
(102,105)(109,121)(110,122)(111,123)(112,118)(113,119)(114,120)(115,124)
(116,125)(117,126)(127,130)(128,131)(129,132)(136,148)(137,149)(138,150)
(139,145)(140,146)(141,147)(142,151)(143,152)(144,153)(154,157)(155,158)
(156,159)(163,175)(164,176)(165,177)(166,172)(167,173)(168,174)(169,178)
(170,179)(171,180)(181,184)(182,185)(183,186)(190,202)(191,203)(192,204)
(193,199)(194,200)(195,201)(196,205)(197,206)(198,207)(208,211)(209,212)
(210,213);
s3 := Sym(216)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)
(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)
(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)
(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)
(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)
(213,216);
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, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
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