Polytope of Type {8,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,12703)
Rank : 3
Schlafli Type : {8,12}
Number of vertices, edges, etc : 72, 432, 108
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 6
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}*576a
   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}*64a
   54-fold quotients : {4,4}*32, {8,2}*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 := (  1,109)(  2,111)(  3,110)(  4,130)(  5,132)(  6,131)(  7,126)(  8,125)
(  9,124)( 10,127)( 11,129)( 12,128)( 13,121)( 14,123)( 15,122)( 16,117)
( 17,116)( 18,115)( 19,118)( 20,120)( 21,119)( 22,112)( 23,114)( 24,113)
( 25,135)( 26,134)( 27,133)( 28,136)( 29,138)( 30,137)( 31,157)( 32,159)
( 33,158)( 34,153)( 35,152)( 36,151)( 37,154)( 38,156)( 39,155)( 40,148)
( 41,150)( 42,149)( 43,144)( 44,143)( 45,142)( 46,145)( 47,147)( 48,146)
( 49,139)( 50,141)( 51,140)( 52,162)( 53,161)( 54,160)( 55,190)( 56,192)
( 57,191)( 58,211)( 59,213)( 60,212)( 61,207)( 62,206)( 63,205)( 64,208)
( 65,210)( 66,209)( 67,202)( 68,204)( 69,203)( 70,198)( 71,197)( 72,196)
( 73,199)( 74,201)( 75,200)( 76,193)( 77,195)( 78,194)( 79,216)( 80,215)
( 81,214)( 82,163)( 83,165)( 84,164)( 85,184)( 86,186)( 87,185)( 88,180)
( 89,179)( 90,178)( 91,181)( 92,183)( 93,182)( 94,175)( 95,177)( 96,176)
( 97,171)( 98,170)( 99,169)(100,172)(101,174)(102,173)(103,166)(104,168)
(105,167)(106,189)(107,188)(108,187);;
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, 13)(  2, 15)(  3, 14)(  4, 21)(  5, 20)(  6, 19)(  8,  9)( 10, 23)
( 11, 22)( 12, 24)( 16, 17)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 48)
( 32, 47)( 33, 46)( 35, 36)( 37, 50)( 38, 49)( 39, 51)( 43, 44)( 52, 54)
( 55, 67)( 56, 69)( 57, 68)( 58, 75)( 59, 74)( 60, 73)( 62, 63)( 64, 77)
( 65, 76)( 66, 78)( 70, 71)( 79, 81)( 82, 94)( 83, 96)( 84, 95)( 85,102)
( 86,101)( 87,100)( 89, 90)( 91,104)( 92,103)( 93,105)( 97, 98)(106,108)
(109,121)(110,123)(111,122)(112,129)(113,128)(114,127)(116,117)(118,131)
(119,130)(120,132)(124,125)(133,135)(136,148)(137,150)(138,149)(139,156)
(140,155)(141,154)(143,144)(145,158)(146,157)(147,159)(151,152)(160,162)
(163,175)(164,177)(165,176)(166,183)(167,182)(168,181)(170,171)(172,185)
(173,184)(174,186)(178,179)(187,189)(190,202)(191,204)(192,203)(193,210)
(194,209)(195,208)(197,198)(199,212)(200,211)(201,213)(205,206)(214,216);;
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, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope