Polytope of Type {12,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,8}*1728c
if this polytope has a name.
Group : SmallGroup(1728,12713)
Rank : 3
Schlafli Type : {12,8}
Number of vertices, edges, etc : 108, 432, 72
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {12,4}*864a
   3-fold quotients : {4,8}*576b
   4-fold quotients : {12,4}*432a
   6-fold quotients : {4,4}*288
   8-fold quotients : {12,4}*216
   12-fold quotients : {4,4}*144
   24-fold quotients : {4,4}*72
   27-fold quotients : {4,8}*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 := (  2,  3)(  4, 22)(  5, 24)(  6, 23)(  7, 18)(  8, 17)(  9, 16)( 10, 19)
( 11, 21)( 12, 20)( 14, 15)( 25, 27)( 29, 30)( 31, 49)( 32, 51)( 33, 50)
( 34, 45)( 35, 44)( 36, 43)( 37, 46)( 38, 48)( 39, 47)( 41, 42)( 52, 54)
( 55, 82)( 56, 84)( 57, 83)( 58,103)( 59,105)( 60,104)( 61, 99)( 62, 98)
( 63, 97)( 64,100)( 65,102)( 66,101)( 67, 94)( 68, 96)( 69, 95)( 70, 90)
( 71, 89)( 72, 88)( 73, 91)( 74, 93)( 75, 92)( 76, 85)( 77, 87)( 78, 86)
( 79,108)( 80,107)( 81,106)(110,111)(112,130)(113,132)(114,131)(115,126)
(116,125)(117,124)(118,127)(119,129)(120,128)(122,123)(133,135)(137,138)
(139,157)(140,159)(141,158)(142,153)(143,152)(144,151)(145,154)(146,156)
(147,155)(149,150)(160,162)(163,190)(164,192)(165,191)(166,211)(167,213)
(168,212)(169,207)(170,206)(171,205)(172,208)(173,210)(174,209)(175,202)
(176,204)(177,203)(178,198)(179,197)(180,196)(181,199)(182,201)(183,200)
(184,193)(185,195)(186,194)(187,216)(188,215)(189,214);;
s1 := (  1,  2)(  4,  6)(  8,  9)( 10, 26)( 11, 25)( 12, 27)( 13, 21)( 14, 20)
( 15, 19)( 16, 22)( 17, 24)( 18, 23)( 28, 29)( 31, 33)( 35, 36)( 37, 53)
( 38, 52)( 39, 54)( 40, 48)( 41, 47)( 42, 46)( 43, 49)( 44, 51)( 45, 50)
( 55, 83)( 56, 82)( 57, 84)( 58, 87)( 59, 86)( 60, 85)( 61, 88)( 62, 90)
( 63, 89)( 64,107)( 65,106)( 66,108)( 67,102)( 68,101)( 69,100)( 70,103)
( 71,105)( 72,104)( 73, 96)( 74, 95)( 75, 94)( 76, 97)( 77, 99)( 78, 98)
( 79, 92)( 80, 91)( 81, 93)(109,164)(110,163)(111,165)(112,168)(113,167)
(114,166)(115,169)(116,171)(117,170)(118,188)(119,187)(120,189)(121,183)
(122,182)(123,181)(124,184)(125,186)(126,185)(127,177)(128,176)(129,175)
(130,178)(131,180)(132,179)(133,173)(134,172)(135,174)(136,191)(137,190)
(138,192)(139,195)(140,194)(141,193)(142,196)(143,198)(144,197)(145,215)
(146,214)(147,216)(148,210)(149,209)(150,208)(151,211)(152,213)(153,212)
(154,204)(155,203)(156,202)(157,205)(158,207)(159,206)(160,200)(161,199)
(162,201);;
s2 := (  1,148)(  2,150)(  3,149)(  4,156)(  5,155)(  6,154)(  7,142)(  8,144)
(  9,143)( 10,158)( 11,157)( 12,159)( 13,136)( 14,138)( 15,137)( 16,152)
( 17,151)( 18,153)( 19,141)( 20,140)( 21,139)( 22,146)( 23,145)( 24,147)
( 25,162)( 26,161)( 27,160)( 28,121)( 29,123)( 30,122)( 31,129)( 32,128)
( 33,127)( 34,115)( 35,117)( 36,116)( 37,131)( 38,130)( 39,132)( 40,109)
( 41,111)( 42,110)( 43,125)( 44,124)( 45,126)( 46,114)( 47,113)( 48,112)
( 49,119)( 50,118)( 51,120)( 52,135)( 53,134)( 54,133)( 55,202)( 56,204)
( 57,203)( 58,210)( 59,209)( 60,208)( 61,196)( 62,198)( 63,197)( 64,212)
( 65,211)( 66,213)( 67,190)( 68,192)( 69,191)( 70,206)( 71,205)( 72,207)
( 73,195)( 74,194)( 75,193)( 76,200)( 77,199)( 78,201)( 79,216)( 80,215)
( 81,214)( 82,175)( 83,177)( 84,176)( 85,183)( 86,182)( 87,181)( 88,169)
( 89,171)( 90,170)( 91,185)( 92,184)( 93,186)( 94,163)( 95,165)( 96,164)
( 97,179)( 98,178)( 99,180)(100,168)(101,167)(102,166)(103,173)(104,172)
(105,174)(106,189)(107,188)(108,187);;
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*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope