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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}*1728f
if this polytope has a name.
Group : SmallGroup(1728,37593)
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}*864f
   3-fold quotients : {8,6,6}*576a, {24,6,2}*576c
   4-fold quotients : {6,6,6}*432d
   6-fold quotients : {4,6,6}*288a, {12,6,2}*288c
   9-fold quotients : {8,2,6}*192, {8,6,2}*192
   12-fold quotients : {2,6,6}*144a, {6,6,2}*144b
   18-fold quotients : {8,2,3}*96, {4,2,6}*96, {4,6,2}*96a
   24-fold quotients : {6,3,2}*72
   27-fold quotients : {8,2,2}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {4,2,2}*32
   72-fold quotients : {2,2,3}*24, {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)
( 61, 88)( 62, 89)( 63, 90)( 64,100)( 65,101)( 66,102)( 67,103)( 68,104)
( 69,105)( 70,106)( 71,107)( 72,108)( 73, 91)( 74, 92)( 75, 93)( 76, 94)
( 77, 95)( 78, 96)( 79, 97)( 80, 98)( 81, 99)(109,163)(110,164)(111,165)
(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,181)(119,182)
(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,172)
(128,173)(129,174)(130,175)(131,176)(132,177)(133,178)(134,179)(135,180)
(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)
(144,198)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)
(152,215)(153,216)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)
(160,205)(161,206)(162,207);;
s1 := (  1,118)(  2,120)(  3,119)(  4,121)(  5,123)(  6,122)(  7,124)(  8,126)
(  9,125)( 10,109)( 11,111)( 12,110)( 13,112)( 14,114)( 15,113)( 16,115)
( 17,117)( 18,116)( 19,127)( 20,129)( 21,128)( 22,130)( 23,132)( 24,131)
( 25,133)( 26,135)( 27,134)( 28,145)( 29,147)( 30,146)( 31,148)( 32,150)
( 33,149)( 34,151)( 35,153)( 36,152)( 37,136)( 38,138)( 39,137)( 40,139)
( 41,141)( 42,140)( 43,142)( 44,144)( 45,143)( 46,154)( 47,156)( 48,155)
( 49,157)( 50,159)( 51,158)( 52,160)( 53,162)( 54,161)( 55,199)( 56,201)
( 57,200)( 58,202)( 59,204)( 60,203)( 61,205)( 62,207)( 63,206)( 64,190)
( 65,192)( 66,191)( 67,193)( 68,195)( 69,194)( 70,196)( 71,198)( 72,197)
( 73,208)( 74,210)( 75,209)( 76,211)( 77,213)( 78,212)( 79,214)( 80,216)
( 81,215)( 82,172)( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)( 88,178)
( 89,180)( 90,179)( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)( 96,167)
( 97,169)( 98,171)( 99,170)(100,181)(101,183)(102,182)(103,184)(104,186)
(105,185)(106,187)(107,189)(108,188);;
s2 := (  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 20)( 11, 19)( 12, 21)( 13, 26)
( 14, 25)( 15, 27)( 16, 23)( 17, 22)( 18, 24)( 28, 29)( 31, 35)( 32, 34)
( 33, 36)( 37, 47)( 38, 46)( 39, 48)( 40, 53)( 41, 52)( 42, 54)( 43, 50)
( 44, 49)( 45, 51)( 55, 56)( 58, 62)( 59, 61)( 60, 63)( 64, 74)( 65, 73)
( 66, 75)( 67, 80)( 68, 79)( 69, 81)( 70, 77)( 71, 76)( 72, 78)( 82, 83)
( 85, 89)( 86, 88)( 87, 90)( 91,101)( 92,100)( 93,102)( 94,107)( 95,106)
( 96,108)( 97,104)( 98,103)( 99,105)(109,110)(112,116)(113,115)(114,117)
(118,128)(119,127)(120,129)(121,134)(122,133)(123,135)(124,131)(125,130)
(126,132)(136,137)(139,143)(140,142)(141,144)(145,155)(146,154)(147,156)
(148,161)(149,160)(150,162)(151,158)(152,157)(153,159)(163,164)(166,170)
(167,169)(168,171)(172,182)(173,181)(174,183)(175,188)(176,187)(177,189)
(178,185)(179,184)(180,186)(190,191)(193,197)(194,196)(195,198)(199,209)
(200,208)(201,210)(202,215)(203,214)(204,216)(205,212)(206,211)(207,213);;
s3 := (  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)( 20, 23)
( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)( 46, 49)
( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)
( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)( 92, 95)
( 93, 96)(100,103)(101,104)(102,105)(109,112)(110,113)(111,114)(118,121)
(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)(138,141)
(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(163,166)(164,167)
(165,168)(172,175)(173,176)(174,177)(181,184)(182,185)(183,186)(190,193)
(191,194)(192,195)(199,202)(200,203)(201,204)(208,211)(209,212)(210,213);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(216)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)
( 60, 87)( 61, 88)( 62, 89)( 63, 90)( 64,100)( 65,101)( 66,102)( 67,103)
( 68,104)( 69,105)( 70,106)( 71,107)( 72,108)( 73, 91)( 74, 92)( 75, 93)
( 76, 94)( 77, 95)( 78, 96)( 79, 97)( 80, 98)( 81, 99)(109,163)(110,164)
(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,181)
(119,182)(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)
(127,172)(128,173)(129,174)(130,175)(131,176)(132,177)(133,178)(134,179)
(135,180)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)
(143,197)(144,198)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)
(151,214)(152,215)(153,216)(154,199)(155,200)(156,201)(157,202)(158,203)
(159,204)(160,205)(161,206)(162,207);
s1 := Sym(216)!(  1,118)(  2,120)(  3,119)(  4,121)(  5,123)(  6,122)(  7,124)
(  8,126)(  9,125)( 10,109)( 11,111)( 12,110)( 13,112)( 14,114)( 15,113)
( 16,115)( 17,117)( 18,116)( 19,127)( 20,129)( 21,128)( 22,130)( 23,132)
( 24,131)( 25,133)( 26,135)( 27,134)( 28,145)( 29,147)( 30,146)( 31,148)
( 32,150)( 33,149)( 34,151)( 35,153)( 36,152)( 37,136)( 38,138)( 39,137)
( 40,139)( 41,141)( 42,140)( 43,142)( 44,144)( 45,143)( 46,154)( 47,156)
( 48,155)( 49,157)( 50,159)( 51,158)( 52,160)( 53,162)( 54,161)( 55,199)
( 56,201)( 57,200)( 58,202)( 59,204)( 60,203)( 61,205)( 62,207)( 63,206)
( 64,190)( 65,192)( 66,191)( 67,193)( 68,195)( 69,194)( 70,196)( 71,198)
( 72,197)( 73,208)( 74,210)( 75,209)( 76,211)( 77,213)( 78,212)( 79,214)
( 80,216)( 81,215)( 82,172)( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)
( 88,178)( 89,180)( 90,179)( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)
( 96,167)( 97,169)( 98,171)( 99,170)(100,181)(101,183)(102,182)(103,184)
(104,186)(105,185)(106,187)(107,189)(108,188);
s2 := Sym(216)!(  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 20)( 11, 19)( 12, 21)
( 13, 26)( 14, 25)( 15, 27)( 16, 23)( 17, 22)( 18, 24)( 28, 29)( 31, 35)
( 32, 34)( 33, 36)( 37, 47)( 38, 46)( 39, 48)( 40, 53)( 41, 52)( 42, 54)
( 43, 50)( 44, 49)( 45, 51)( 55, 56)( 58, 62)( 59, 61)( 60, 63)( 64, 74)
( 65, 73)( 66, 75)( 67, 80)( 68, 79)( 69, 81)( 70, 77)( 71, 76)( 72, 78)
( 82, 83)( 85, 89)( 86, 88)( 87, 90)( 91,101)( 92,100)( 93,102)( 94,107)
( 95,106)( 96,108)( 97,104)( 98,103)( 99,105)(109,110)(112,116)(113,115)
(114,117)(118,128)(119,127)(120,129)(121,134)(122,133)(123,135)(124,131)
(125,130)(126,132)(136,137)(139,143)(140,142)(141,144)(145,155)(146,154)
(147,156)(148,161)(149,160)(150,162)(151,158)(152,157)(153,159)(163,164)
(166,170)(167,169)(168,171)(172,182)(173,181)(174,183)(175,188)(176,187)
(177,189)(178,185)(179,184)(180,186)(190,191)(193,197)(194,196)(195,198)
(199,209)(200,208)(201,210)(202,215)(203,214)(204,216)(205,212)(206,211)
(207,213);
s3 := Sym(216)!(  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)
( 20, 23)( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)
( 46, 49)( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)
( 66, 69)( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)
( 92, 95)( 93, 96)(100,103)(101,104)(102,105)(109,112)(110,113)(111,114)
(118,121)(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)
(138,141)(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(163,166)
(164,167)(165,168)(172,175)(173,176)(174,177)(181,184)(182,185)(183,186)
(190,193)(191,194)(192,195)(199,202)(200,203)(201,204)(208,211)(209,212)
(210,213);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope