Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1728e
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 24, 72, 72, 6
Order of s₀s₁s₂s₃ : 12
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 : {6,3,6}*864b
   3-fold quotients : {6,6,2}*576a
   4-fold quotients : {6,6,6}*432f
   6-fold quotients : {6,3,2}*288
   8-fold quotients : {6,3,6}*216
   9-fold quotients : {6,6,2}*192
   12-fold quotients : {2,6,6}*144c, {6,6,2}*144b
   18-fold quotients : {3,6,2}*96, {6,3,2}*96
   24-fold quotients : {2,3,6}*72, {6,3,2}*72
   36-fold quotients : {3,3,2}*48, {2,6,2}*48
   72-fold quotients : {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope