Polytope of Type {8,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,15977)
Rank : 4
Schlafli Type : {8,6,6}
Number of vertices, edges, etc : 8, 72, 54, 18
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 : {4,6,6}*864a
   3-fold quotients : {8,6,6}*576b
   4-fold quotients : {2,6,6}*432c
   6-fold quotients : {4,6,6}*288b
   8-fold quotients : {2,3,6}*216
   9-fold quotients : {8,6,2}*192
   12-fold quotients : {2,6,6}*144c
   18-fold quotients : {4,6,2}*96a
   24-fold quotients : {2,3,6}*72
   27-fold quotients : {8,2,2}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {4,2,2}*32
   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 := ( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)
( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)
( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)
( 79,106)( 80,107)( 81,108)(109,163)(110,164)(111,165)(112,166)(113,167)
(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)
(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)
(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)
(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)
(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)
(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)
(162,216);;
s1 := (  1,109)(  2,110)(  3,111)(  4,115)(  5,116)(  6,117)(  7,112)(  8,113)
(  9,114)( 10,127)( 11,128)( 12,129)( 13,133)( 14,134)( 15,135)( 16,130)
( 17,131)( 18,132)( 19,118)( 20,119)( 21,120)( 22,124)( 23,125)( 24,126)
( 25,121)( 26,122)( 27,123)( 28,136)( 29,137)( 30,138)( 31,142)( 32,143)
( 33,144)( 34,139)( 35,140)( 36,141)( 37,154)( 38,155)( 39,156)( 40,160)
( 41,161)( 42,162)( 43,157)( 44,158)( 45,159)( 46,145)( 47,146)( 48,147)
( 49,151)( 50,152)( 51,153)( 52,148)( 53,149)( 54,150)( 55,190)( 56,191)
( 57,192)( 58,196)( 59,197)( 60,198)( 61,193)( 62,194)( 63,195)( 64,208)
( 65,209)( 66,210)( 67,214)( 68,215)( 69,216)( 70,211)( 71,212)( 72,213)
( 73,199)( 74,200)( 75,201)( 76,205)( 77,206)( 78,207)( 79,202)( 80,203)
( 81,204)( 82,163)( 83,164)( 84,165)( 85,169)( 86,170)( 87,171)( 88,166)
( 89,167)( 90,168)( 91,181)( 92,182)( 93,183)( 94,187)( 95,188)( 96,189)
( 97,184)( 98,185)( 99,186)(100,172)(101,173)(102,174)(103,178)(104,179)
(105,180)(106,175)(107,176)(108,177);;
s2 := (  1, 23)(  2, 24)(  3, 22)(  4, 21)(  5, 19)(  6, 20)(  7, 25)(  8, 26)
(  9, 27)( 10, 14)( 11, 15)( 12, 13)( 28, 50)( 29, 51)( 30, 49)( 31, 48)
( 32, 46)( 33, 47)( 34, 52)( 35, 53)( 36, 54)( 37, 41)( 38, 42)( 39, 40)
( 55, 77)( 56, 78)( 57, 76)( 58, 75)( 59, 73)( 60, 74)( 61, 79)( 62, 80)
( 63, 81)( 64, 68)( 65, 69)( 66, 67)( 82,104)( 83,105)( 84,103)( 85,102)
( 86,100)( 87,101)( 88,106)( 89,107)( 90,108)( 91, 95)( 92, 96)( 93, 94)
(109,131)(110,132)(111,130)(112,129)(113,127)(114,128)(115,133)(116,134)
(117,135)(118,122)(119,123)(120,121)(136,158)(137,159)(138,157)(139,156)
(140,154)(141,155)(142,160)(143,161)(144,162)(145,149)(146,150)(147,148)
(163,185)(164,186)(165,184)(166,183)(167,181)(168,182)(169,187)(170,188)
(171,189)(172,176)(173,177)(174,175)(190,212)(191,213)(192,211)(193,210)
(194,208)(195,209)(196,214)(197,215)(198,216)(199,203)(200,204)(201,202);;
s3 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)
(164,165)(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)(176,186)
(177,185)(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)(199,208)
(200,210)(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)(207,215);;
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, 
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope