Polytope of Type {6,6,6}

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