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Polytope of Type {8,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,12703)
Rank : 4
Schlafli Type : {8,12,6}
Number of vertices, edges, etc : 8, 72, 54, 9
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Non-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 : {4,12,6}*864b
   3-fold quotients : {8,4,6}*576
   4-fold quotients : {2,12,6}*432b
   6-fold quotients : {4,4,6}*288
   12-fold quotients : {2,4,6}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)(  8,116)
(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)( 56,191)
( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)( 64,199)
( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)( 72,207)
( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)( 80,215)
( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)( 88,169)
( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)( 96,177)
( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)(104,185)
(105,186)(106,187)(107,188)(108,189);;
s1 := (  2,  3)(  4,  5)(  7,  9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)( 14, 19)
( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)( 37, 52)
( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)( 45, 49)
( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)( 62, 89)
( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)( 70,105)
( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)( 78, 97)
( 79, 91)( 80, 93)( 81, 92)(109,163)(110,165)(111,164)(112,167)(113,166)
(114,168)(115,171)(116,170)(117,169)(118,187)(119,189)(120,188)(121,182)
(122,181)(123,183)(124,186)(125,185)(126,184)(127,176)(128,175)(129,177)
(130,180)(131,179)(132,178)(133,172)(134,174)(135,173)(136,190)(137,192)
(138,191)(139,194)(140,193)(141,195)(142,198)(143,197)(144,196)(145,214)
(146,216)(147,215)(148,209)(149,208)(150,210)(151,213)(152,212)(153,211)
(154,203)(155,202)(156,204)(157,207)(158,206)(159,205)(160,199)(161,201)
(162,200);;
s2 := (  1,  2)(  4, 17)(  5, 16)(  6, 18)(  7, 22)(  8, 24)(  9, 23)( 10, 12)
( 13, 27)( 14, 26)( 15, 25)( 20, 21)( 28, 29)( 31, 44)( 32, 43)( 33, 45)
( 34, 49)( 35, 51)( 36, 50)( 37, 39)( 40, 54)( 41, 53)( 42, 52)( 47, 48)
( 55, 56)( 58, 71)( 59, 70)( 60, 72)( 61, 76)( 62, 78)( 63, 77)( 64, 66)
( 67, 81)( 68, 80)( 69, 79)( 74, 75)( 82, 83)( 85, 98)( 86, 97)( 87, 99)
( 88,103)( 89,105)( 90,104)( 91, 93)( 94,108)( 95,107)( 96,106)(101,102)
(109,110)(112,125)(113,124)(114,126)(115,130)(116,132)(117,131)(118,120)
(121,135)(122,134)(123,133)(128,129)(136,137)(139,152)(140,151)(141,153)
(142,157)(143,159)(144,158)(145,147)(148,162)(149,161)(150,160)(155,156)
(163,164)(166,179)(167,178)(168,180)(169,184)(170,186)(171,185)(172,174)
(175,189)(176,188)(177,187)(182,183)(190,191)(193,206)(194,205)(195,207)
(196,211)(197,213)(198,212)(199,201)(202,216)(203,215)(204,214)(209,210);;
s3 := (  1,  8)(  2,  9)(  3,  7)( 10, 25)( 11, 26)( 12, 27)( 13, 24)( 14, 22)
( 15, 23)( 16, 20)( 17, 21)( 18, 19)( 28, 35)( 29, 36)( 30, 34)( 37, 52)
( 38, 53)( 39, 54)( 40, 51)( 41, 49)( 42, 50)( 43, 47)( 44, 48)( 45, 46)
( 55, 62)( 56, 63)( 57, 61)( 64, 79)( 65, 80)( 66, 81)( 67, 78)( 68, 76)
( 69, 77)( 70, 74)( 71, 75)( 72, 73)( 82, 89)( 83, 90)( 84, 88)( 91,106)
( 92,107)( 93,108)( 94,105)( 95,103)( 96,104)( 97,101)( 98,102)( 99,100)
(109,116)(110,117)(111,115)(118,133)(119,134)(120,135)(121,132)(122,130)
(123,131)(124,128)(125,129)(126,127)(136,143)(137,144)(138,142)(145,160)
(146,161)(147,162)(148,159)(149,157)(150,158)(151,155)(152,156)(153,154)
(163,170)(164,171)(165,169)(172,187)(173,188)(174,189)(175,186)(176,184)
(177,185)(178,182)(179,183)(180,181)(190,197)(191,198)(192,196)(199,214)
(200,215)(201,216)(202,213)(203,211)(204,212)(205,209)(206,210)(207,208);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(216)!(  1,109)(  2,110)(  3,111)(  4,112)(  5,113)(  6,114)(  7,115)
(  8,116)(  9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)
( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)
( 24,132)( 25,133)( 26,134)( 27,135)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,190)
( 56,191)( 57,192)( 58,193)( 59,194)( 60,195)( 61,196)( 62,197)( 63,198)
( 64,199)( 65,200)( 66,201)( 67,202)( 68,203)( 69,204)( 70,205)( 71,206)
( 72,207)( 73,208)( 74,209)( 75,210)( 76,211)( 77,212)( 78,213)( 79,214)
( 80,215)( 81,216)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)( 87,168)
( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)( 95,176)
( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)(103,184)
(104,185)(105,186)(106,187)(107,188)(108,189);
s1 := Sym(216)!(  2,  3)(  4,  5)(  7,  9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)
( 14, 19)( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)
( 37, 52)( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)
( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)
( 62, 89)( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)
( 70,105)( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)
( 78, 97)( 79, 91)( 80, 93)( 81, 92)(109,163)(110,165)(111,164)(112,167)
(113,166)(114,168)(115,171)(116,170)(117,169)(118,187)(119,189)(120,188)
(121,182)(122,181)(123,183)(124,186)(125,185)(126,184)(127,176)(128,175)
(129,177)(130,180)(131,179)(132,178)(133,172)(134,174)(135,173)(136,190)
(137,192)(138,191)(139,194)(140,193)(141,195)(142,198)(143,197)(144,196)
(145,214)(146,216)(147,215)(148,209)(149,208)(150,210)(151,213)(152,212)
(153,211)(154,203)(155,202)(156,204)(157,207)(158,206)(159,205)(160,199)
(161,201)(162,200);
s2 := Sym(216)!(  1,  2)(  4, 17)(  5, 16)(  6, 18)(  7, 22)(  8, 24)(  9, 23)
( 10, 12)( 13, 27)( 14, 26)( 15, 25)( 20, 21)( 28, 29)( 31, 44)( 32, 43)
( 33, 45)( 34, 49)( 35, 51)( 36, 50)( 37, 39)( 40, 54)( 41, 53)( 42, 52)
( 47, 48)( 55, 56)( 58, 71)( 59, 70)( 60, 72)( 61, 76)( 62, 78)( 63, 77)
( 64, 66)( 67, 81)( 68, 80)( 69, 79)( 74, 75)( 82, 83)( 85, 98)( 86, 97)
( 87, 99)( 88,103)( 89,105)( 90,104)( 91, 93)( 94,108)( 95,107)( 96,106)
(101,102)(109,110)(112,125)(113,124)(114,126)(115,130)(116,132)(117,131)
(118,120)(121,135)(122,134)(123,133)(128,129)(136,137)(139,152)(140,151)
(141,153)(142,157)(143,159)(144,158)(145,147)(148,162)(149,161)(150,160)
(155,156)(163,164)(166,179)(167,178)(168,180)(169,184)(170,186)(171,185)
(172,174)(175,189)(176,188)(177,187)(182,183)(190,191)(193,206)(194,205)
(195,207)(196,211)(197,213)(198,212)(199,201)(202,216)(203,215)(204,214)
(209,210);
s3 := Sym(216)!(  1,  8)(  2,  9)(  3,  7)( 10, 25)( 11, 26)( 12, 27)( 13, 24)
( 14, 22)( 15, 23)( 16, 20)( 17, 21)( 18, 19)( 28, 35)( 29, 36)( 30, 34)
( 37, 52)( 38, 53)( 39, 54)( 40, 51)( 41, 49)( 42, 50)( 43, 47)( 44, 48)
( 45, 46)( 55, 62)( 56, 63)( 57, 61)( 64, 79)( 65, 80)( 66, 81)( 67, 78)
( 68, 76)( 69, 77)( 70, 74)( 71, 75)( 72, 73)( 82, 89)( 83, 90)( 84, 88)
( 91,106)( 92,107)( 93,108)( 94,105)( 95,103)( 96,104)( 97,101)( 98,102)
( 99,100)(109,116)(110,117)(111,115)(118,133)(119,134)(120,135)(121,132)
(122,130)(123,131)(124,128)(125,129)(126,127)(136,143)(137,144)(138,142)
(145,160)(146,161)(147,162)(148,159)(149,157)(150,158)(151,155)(152,156)
(153,154)(163,170)(164,171)(165,169)(172,187)(173,188)(174,189)(175,186)
(176,184)(177,185)(178,182)(179,183)(180,181)(190,197)(191,198)(192,196)
(199,214)(200,215)(201,216)(202,213)(203,211)(204,212)(205,209)(206,210)
(207,208);
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*s2*s3*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
to this polytope