Polytope of Type {8,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,12713)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 72, 432, 108
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,12}*864a
   3-fold quotients : {8,4}*576b
   4-fold quotients : {4,12}*432a
   6-fold quotients : {4,4}*288
   8-fold quotients : {4,12}*216
   12-fold quotients : {4,4}*144
   24-fold quotients : {4,4}*72
   27-fold quotients : {8,4}*64b
   54-fold quotients : {4,4}*32
   108-fold quotients : {2,4}*16, {4,2}*16
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1, 
s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1, 
s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s0 >;

References : None.
to this polytope