Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 6, 72, 72, 24
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}*864a
   3-fold quotients : {2,6,6}*576b
   4-fold quotients : {6,6,6}*432f
   6-fold quotients : {2,3,6}*288
   8-fold quotients : {6,3,6}*216
   9-fold quotients : {2,6,6}*192
   12-fold quotients : {2,6,6}*144c, {6,6,2}*144b
   18-fold quotients : {2,3,6}*96, {2,6,3}*96
   24-fold quotients : {2,3,6}*72, {6,3,2}*72
   36-fold quotients : {2,3,3}*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 := ( 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);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)(  8, 23)
(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 27, 28)( 29, 33)( 30, 34)( 31, 36)
( 32, 35)( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 93)( 42, 94)( 43, 96)
( 44, 95)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 73)( 50, 74)( 51, 76)
( 52, 75)( 53, 81)( 54, 82)( 55, 84)( 56, 83)( 57, 77)( 58, 78)( 59, 80)
( 60, 79)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,105)( 66,106)( 67,108)
( 68,107)( 69,101)( 70,102)( 71,104)( 72,103)(109,121)(110,122)(111,124)
(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)(119,128)
(120,127)(135,136)(137,141)(138,142)(139,144)(140,143)(145,193)(146,194)
(147,196)(148,195)(149,201)(150,202)(151,204)(152,203)(153,197)(154,198)
(155,200)(156,199)(157,181)(158,182)(159,184)(160,183)(161,189)(162,190)
(163,192)(164,191)(165,185)(166,186)(167,188)(168,187)(169,205)(170,206)
(171,208)(172,207)(173,213)(174,214)(175,216)(176,215)(177,209)(178,210)
(179,212)(180,211);;
s2 := (  1,149)(  2,152)(  3,151)(  4,150)(  5,145)(  6,148)(  7,147)(  8,146)
(  9,153)( 10,156)( 11,155)( 12,154)( 13,173)( 14,176)( 15,175)( 16,174)
( 17,169)( 18,172)( 19,171)( 20,170)( 21,177)( 22,180)( 23,179)( 24,178)
( 25,161)( 26,164)( 27,163)( 28,162)( 29,157)( 30,160)( 31,159)( 32,158)
( 33,165)( 34,168)( 35,167)( 36,166)( 37,113)( 38,116)( 39,115)( 40,114)
( 41,109)( 42,112)( 43,111)( 44,110)( 45,117)( 46,120)( 47,119)( 48,118)
( 49,137)( 50,140)( 51,139)( 52,138)( 53,133)( 54,136)( 55,135)( 56,134)
( 57,141)( 58,144)( 59,143)( 60,142)( 61,125)( 62,128)( 63,127)( 64,126)
( 65,121)( 66,124)( 67,123)( 68,122)( 69,129)( 70,132)( 71,131)( 72,130)
( 73,185)( 74,188)( 75,187)( 76,186)( 77,181)( 78,184)( 79,183)( 80,182)
( 81,189)( 82,192)( 83,191)( 84,190)( 85,209)( 86,212)( 87,211)( 88,210)
( 89,205)( 90,208)( 91,207)( 92,206)( 93,213)( 94,216)( 95,215)( 96,214)
( 97,197)( 98,200)( 99,199)(100,198)(101,193)(102,196)(103,195)(104,194)
(105,201)(106,204)(107,203)(108,202);;
s3 := (  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)( 29, 30)
( 33, 34)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 79)
( 44, 80)( 45, 82)( 46, 81)( 47, 83)( 48, 84)( 49, 86)( 50, 85)( 51, 87)
( 52, 88)( 53, 90)( 54, 89)( 55, 91)( 56, 92)( 57, 94)( 58, 93)( 59, 95)
( 60, 96)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,102)( 66,101)( 67,103)
( 68,104)( 69,106)( 70,105)( 71,107)( 72,108)(109,110)(113,114)(117,118)
(121,122)(125,126)(129,130)(133,134)(137,138)(141,142)(145,182)(146,181)
(147,183)(148,184)(149,186)(150,185)(151,187)(152,188)(153,190)(154,189)
(155,191)(156,192)(157,194)(158,193)(159,195)(160,196)(161,198)(162,197)
(163,199)(164,200)(165,202)(166,201)(167,203)(168,204)(169,206)(170,205)
(171,207)(172,208)(173,210)(174,209)(175,211)(176,212)(177,214)(178,213)
(179,215)(180,216);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope