Polytope of Type {6,24,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24,6}*1728d
if this polytope has a name.
Group : SmallGroup(1728,33799)
Rank : 4
Schlafli Type : {6,24,6}
Number of vertices, edges, etc : 6, 72, 72, 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 : {6,12,6}*864d
   3-fold quotients : {2,24,6}*576b, {6,24,2}*576a
   4-fold quotients : {6,6,6}*432e
   6-fold quotients : {2,12,6}*288b, {6,12,2}*288a
   9-fold quotients : {2,24,2}*192, {6,8,2}*192
   12-fold quotients : {2,6,6}*144c, {6,6,2}*144a
   18-fold quotients : {2,12,2}*96, {6,4,2}*96a
   24-fold quotients : {2,3,6}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,6,2}*48, {6,2,2}*48
   54-fold quotients : {2,4,2}*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 := (  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);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 22)( 11, 24)( 12, 23)( 13, 19)
( 14, 21)( 15, 20)( 16, 25)( 17, 27)( 18, 26)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 52)
( 44, 54)( 45, 53)( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)
( 61, 88)( 62, 90)( 63, 89)( 64,103)( 65,105)( 66,104)( 67,100)( 68,102)
( 69,101)( 70,106)( 71,108)( 72,107)( 73, 94)( 74, 96)( 75, 95)( 76, 91)
( 77, 93)( 78, 92)( 79, 97)( 80, 99)( 81, 98)(109,166)(110,168)(111,167)
(112,163)(113,165)(114,164)(115,169)(116,171)(117,170)(118,184)(119,186)
(120,185)(121,181)(122,183)(123,182)(124,187)(125,189)(126,188)(127,175)
(128,177)(129,176)(130,172)(131,174)(132,173)(133,178)(134,180)(135,179)
(136,193)(137,195)(138,194)(139,190)(140,192)(141,191)(142,196)(143,198)
(144,197)(145,211)(146,213)(147,212)(148,208)(149,210)(150,209)(151,214)
(152,216)(153,215)(154,202)(155,204)(156,203)(157,199)(158,201)(159,200)
(160,205)(161,207)(162,206);;
s2 := (  1,119)(  2,118)(  3,120)(  4,122)(  5,121)(  6,123)(  7,125)(  8,124)
(  9,126)( 10,110)( 11,109)( 12,111)( 13,113)( 14,112)( 15,114)( 16,116)
( 17,115)( 18,117)( 19,128)( 20,127)( 21,129)( 22,131)( 23,130)( 24,132)
( 25,134)( 26,133)( 27,135)( 28,146)( 29,145)( 30,147)( 31,149)( 32,148)
( 33,150)( 34,152)( 35,151)( 36,153)( 37,137)( 38,136)( 39,138)( 40,140)
( 41,139)( 42,141)( 43,143)( 44,142)( 45,144)( 46,155)( 47,154)( 48,156)
( 49,158)( 50,157)( 51,159)( 52,161)( 53,160)( 54,162)( 55,200)( 56,199)
( 57,201)( 58,203)( 59,202)( 60,204)( 61,206)( 62,205)( 63,207)( 64,191)
( 65,190)( 66,192)( 67,194)( 68,193)( 69,195)( 70,197)( 71,196)( 72,198)
( 73,209)( 74,208)( 75,210)( 76,212)( 77,211)( 78,213)( 79,215)( 80,214)
( 81,216)( 82,173)( 83,172)( 84,174)( 85,176)( 86,175)( 87,177)( 88,179)
( 89,178)( 90,180)( 91,164)( 92,163)( 93,165)( 94,167)( 95,166)( 96,168)
( 97,170)( 98,169)( 99,171)(100,182)(101,181)(102,183)(103,185)(104,184)
(105,186)(106,188)(107,187)(108,189);;
s3 := ( 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)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)(173,182)
(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)(199,208)
(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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*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)!(  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);
s1 := Sym(216)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 22)( 11, 24)( 12, 23)
( 13, 19)( 14, 21)( 15, 20)( 16, 25)( 17, 27)( 18, 26)( 28, 31)( 29, 33)
( 30, 32)( 35, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)
( 43, 52)( 44, 54)( 45, 53)( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)
( 60, 83)( 61, 88)( 62, 90)( 63, 89)( 64,103)( 65,105)( 66,104)( 67,100)
( 68,102)( 69,101)( 70,106)( 71,108)( 72,107)( 73, 94)( 74, 96)( 75, 95)
( 76, 91)( 77, 93)( 78, 92)( 79, 97)( 80, 99)( 81, 98)(109,166)(110,168)
(111,167)(112,163)(113,165)(114,164)(115,169)(116,171)(117,170)(118,184)
(119,186)(120,185)(121,181)(122,183)(123,182)(124,187)(125,189)(126,188)
(127,175)(128,177)(129,176)(130,172)(131,174)(132,173)(133,178)(134,180)
(135,179)(136,193)(137,195)(138,194)(139,190)(140,192)(141,191)(142,196)
(143,198)(144,197)(145,211)(146,213)(147,212)(148,208)(149,210)(150,209)
(151,214)(152,216)(153,215)(154,202)(155,204)(156,203)(157,199)(158,201)
(159,200)(160,205)(161,207)(162,206);
s2 := Sym(216)!(  1,119)(  2,118)(  3,120)(  4,122)(  5,121)(  6,123)(  7,125)
(  8,124)(  9,126)( 10,110)( 11,109)( 12,111)( 13,113)( 14,112)( 15,114)
( 16,116)( 17,115)( 18,117)( 19,128)( 20,127)( 21,129)( 22,131)( 23,130)
( 24,132)( 25,134)( 26,133)( 27,135)( 28,146)( 29,145)( 30,147)( 31,149)
( 32,148)( 33,150)( 34,152)( 35,151)( 36,153)( 37,137)( 38,136)( 39,138)
( 40,140)( 41,139)( 42,141)( 43,143)( 44,142)( 45,144)( 46,155)( 47,154)
( 48,156)( 49,158)( 50,157)( 51,159)( 52,161)( 53,160)( 54,162)( 55,200)
( 56,199)( 57,201)( 58,203)( 59,202)( 60,204)( 61,206)( 62,205)( 63,207)
( 64,191)( 65,190)( 66,192)( 67,194)( 68,193)( 69,195)( 70,197)( 71,196)
( 72,198)( 73,209)( 74,208)( 75,210)( 76,212)( 77,211)( 78,213)( 79,215)
( 80,214)( 81,216)( 82,173)( 83,172)( 84,174)( 85,176)( 86,175)( 87,177)
( 88,179)( 89,178)( 90,180)( 91,164)( 92,163)( 93,165)( 94,167)( 95,166)
( 96,168)( 97,170)( 98,169)( 99,171)(100,182)(101,181)(102,183)(103,185)
(104,184)(105,186)(106,188)(107,187)(108,189);
s3 := 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)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)
(173,182)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)
(199,208)(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)
(207,216);
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope